Credit: The team list is taken from the guide written on TrueAchievements by user EmporerDragon. I feel that the explanation they included is not clear enough and the correction given by user Blue Radium gave didn't feel clear enough either, though it certainly helped. So this is my version of their explanations using the provided team list.
(Also, apologies if it bothers you: I use caps for emphasis because I first wrote this in basic text for myself before deciding to share it and am too lazy to convert anything to bold now)
Small note: I recommend waiting until you have at least gotten to the point where you have Chapter 5 available. Even better if you have beaten World's End to unlock all but the last four/five Other 2 characters. Best if you have unlocked all characters (Kyubi and Hundun are likely the last two you will unlock in Story Mode).
(My examples use the first listed group)
1.1) Pick each team in this order: Starting from left to right, select those team members in the same order in the Team Selector. (Example, from first to last and shifting right each time: Xiahou Yuan, then Xiahou Dun, then Xiahou Ba)
1.2) STAY on the LAST member and confirm the team, setting that person as the Team Leader (example: Xiahou Ba).
1.note) When you see the team list in the Camp Menu, they will appear in seeming reverse order. This feels weird, but if that's what you see then you set them correctly. (Example, from left to right: Xiahou Ba, Xiahou Dun, Xiahou Yuan)
2) Pick any stage on any difficulty (I suggest either one you haven't already beaten but intend to do, for the sake of saving time, or any stage that is quick to beat and on Easy due to another hidden requirement)
3.1) COMBO TIME! Find any enemy that can be harmed (this should not be hard).
3.2) Initiate any Charge Attack using your Team Leader (Example: Xiahou Ba)
3.3) While that Charge Attack is in action, start to Swap Character using the Right-side button (for Xbox-style controller this is the Right Trigger) and spam that button!
3.4) Keep using the Swap Character action to carry out the cool team combo until you get BACK to the Team Leader as they do a cool finisher. (If you keep pressing the button and continue swapping afterwards, there is no penalty other than looking silly swapping characters in place and maybe taking a couple hits.)
3.5) If you didn't defeat at least one enemy (preferably an Officer to be really certain, but I don't think that is required), then do it again until you are absolutely positive you have defeated at least one enemy using the special Triple Rush. Doing this several times increases the odds of the game not messing up and failing to track a successful combo anyways, so it's a good idea to do this a bunch of times to be safe.
4) FINISH THE LEVEL AND SAVE THE GAME. Be sure to save after every single attempt and be sure to track your progress. (I use a Google Sheet with a checkbox next to each team, and only mark a team after saving the game with that team having returned to camp successful in defeating enemies with their combo.)
NOTE: You MUST defeat at least one enemy using a special combo for that team's combo to count towards progressing the achievement. You can attempt this multiple times in a level (of course), and I actually recommend you do so and that you kill at least one Officer this way. It seems that the game randomly might not track this every single time you do it so to be safe you should use this to defeat several opponents over several uses of the combo in a stage. By including an Officer, you know for certain that you finished them off using this move.
WARNING: If the team is in the wrong order OR you start the combo with the wrong character, you will get a GENERIC team combo. (Obviously, that does not count for the achievement) You need the special team combo that has to be started by the correct character and done in the correct order.
Easy fix: Just switch to the correct character and do it again. If you put them in the wrong order, then you can use the Left-side Swap Character (Left Trigger) instead of RT.
Now, the Team List
There are 60 teams, and they are not all the possible teams. Some characters even show up in more than one team!
Also, teams DO include post-end-game characters: Hundun (from clearing Gauntlet Mode Stage 4 at least once), and Kyubi (from clearing a Story Mode Chapter 7 Side Mission, which is unlocked from the final battle of Chapter 8) are likely to be the last characters you will unlock.
Note: Lu Bu has a unique unlock requirement of first unlocking Diaochan then putting her on the team and replaying Battle of Osaka Castle to have her meet Lu Bu WITHOUT defeating him. After they finish their dialogue he joins if you finish the stage with him still alive.
The others are either from clearing all regular Story Mode battles (ie: unmissable) or via Side Missions that are unlocked by getting a set of characters to have at least one full bar of Bonds with each other (the needed bonds are the characters mentioned in the Side Stage's description while it is grayed out. This will likely require doing at least one mission with each of those members on a team because the Tea House refuses to do that last +1 point needed to finish Bonds bar. If you have a Bonds Number, when looking at Bonds List, however, you can instantly gain one bar regardless of battles. I don't know how that is earned, but it really saves time when you have it and it can be earned from either side of a pairing of Bonds.
1) Xiahou Yuan, Xiahou Dun, Xiahou Ba
2) Cao Ren, Dian Wei, Xu Zhu
3) Zhang Liao, Xu Huang, Zhang He
4) Zhou Yu, Taishi Ci, Sun Ce
5) Sun Shangxiang, Sun Ce, Sun Quan
6) Gan Ning, Ling Tong, Lu Meng
7) Zhang Fei, Guan Yu, Liu Bei
8) Guan Suo, Guan Ping, Guan Yu
9) Ma Chao, Huang Zhong, Zhao Yun
10) Xu Shu, Zhuge Liang, Pang Tong
11) Sima Shi, Sima Zhao, Sima Yi
12) Ma Dai, Pang De, Ma Chao
13) Wei Yan, Meng Huo, Zhurong
14) Lu Xun, Cao Pi, Jiang Wei
15) Zhenji, Yueying, Lianshi
16) Deng Ai, Meng Huo, Huang Gai
17) Keiji Maeda, Toshiie Maeda, Katsuie Shibata
18) Kanetsugu Naoe, Aya, Kenshin Uesugi
19) Yukimura Sanada, Kunoichi, Shingen Takeda
20) Kiyomasa Kato, Masanori Fukushima, Mitsunari Ishida
21) Yoshihiro Shimazu, Ginchiyo Tachibana, Muneshige Tachibana
22) Kenshin Uesugi, Shingen Takeda, Ujiyasu Hojo
23) Nagamasa Azai, Muneshige Tachibana, Hideyoshi Toyotomi
24) Muneshige Tachibana, Tadakatsu Honda, Keiji Maeda
25) Cai Wenji, Zhenji, Motochika Chosokabe
26) Daqiao, Mitsunari Ishida, Xiaoqiao
27) Xingcai, Ina, Guo Jia
28) Daqiao, Himiko, Bao Sanniang
29) Zhuge Dan, Sima Zhao, Sakon Shima
30) Dong Zhuo, Da Ji, Orochi
31) Yoshimoto Imagawa, Zhong Hui, Yuan Shao
32) Meng Huo, Ding Feng, Kotaro Fuma
33) Zhurong, Nene, Wang Yuanji
34) Goemon Ishikawa, Hideyoshi Toyotomi, Ling Tong
35) Zhang Jiao, Magoichi Saika, Gracia
36) Ranmaru Mori, Mitsuhide Akechi, Zhou Tai
37) No, Wang Yi, Oichi
38) Liu Shan, Motochika Chosokabe, Yoshimoto Imagawa
39) Kojiro Sasaki, Musashi Miyamoto, Xu Huang
40) Hanbei Takenaka, Jia Xu, Kanbei Kuroda
41) Hanzo Hattori, Kotaro Fuma, Ryu Hayabusa
42) Da Ji, Motonari Mori, Achilles
43) Kasumi, Kunoichi, Ayane
44) Diaochan, Okuni, Sanzang
45) Lu Bu, Cyborg Nezha, Yoshitsune Minamoto (This marks the first team in the list with post-game characters. Also, Lu Bu with his unique unlock is here, though you already have to have him to get Nezha (cyborg))
46) Zuo Ci, Kanetsugu Naoe, Seimei Abe
47) Nobunaga Oda, Cao Cao, Nemea
48) Liu Bei, Ieyasu Tokugawa, Joan of Arc
49) Guo Huai, Masamune Date, Rachel
50) Sun Shangxiang, Kai, Sterkenburg Cranach
51) Sun Jian, Nene, Sophitia Alexandra
52) Ayane, Rachel, Momiji
53) Fu Xi, Nuwa, Shennong
54) Benkei, Yoshitsune Minamoto, Kiyomori Taira
55) Kiyomori Taira, Orochi X, Sun Wukong
56) Shuten Doji, Taigong Wang, Kaguya
57) Da Ji, Seimei Abe, Mae Tamamo (Teams 57, 58, and 60 both require at least one character from Chapters 5-8)
58) Susano'o, Yinglong, Human Nezha
59) Gyuki, Dodomeki, Hundun (Teams 59 and 60 both require clearing Gauntlet Mode Level 4)
60) Hundun, Kyubi no Kitsune, Orochi X
Training suggestion to save time for the characters you don't care to use:
If you farm in Gauntlet Mode and get one person super strong (personally, Kaguya is always my first to reach ultimate power because she is just that good and available after the first battle in Story Mode), then you can easily get people ready this way (to guarantee getting through the list as fast as possible, even a lv. 1 untrained character will one-shot almost anything via getting a Mystic Weapon, especially if it is a +81 version):
-Mystic Bells: This is the single most important item in the entire game and should be the very first thing you do, even before Story Mode. In Gauntlet Mode, talk to the Mystic and accept the Mission "Brawlers Badge: Defeat a total of 30000 enemy soldiers." This total counts across all Gauntlet Mode Stages and even if you lose (I think). So just repeat DLC Anegawa and get this item ASAP and equip it to absolutely everyone as their first item. You absolutely want this in Story mode regardless! (Missing picking up the Rare Items in Story mode absolutely sucks with their ridiculous KO requirements sometimes being nearly impossible (and being limited to Hard/Chaos), having this equipped guarantees you can't miss it when it drops.)
-Power Training: Gauntlet Mode > DLC Stages > Anegawa (Super-fast power-leveling, due to ALL enemies dropping bonus XP. Even if you aren't strong enough to win and so get defeated, you always keep all XP earned! So this is absolutely a solid strategy even from the beginning of the game with any team. Plus you earn massive amounts of Growth Points to power-level anyone you want.)
-Unlock Characters (for Gauntlet Mode): Gauntlet Mode > DLC Stages > Dongkou (Numerous chests have GUARANTEED NEW Officer Bond Crystals. Also ANY chest on this stage can also have these, so you can very quickly farm the entire game's roster this way. You DO have to finish the stage, so have a setup to adventure quickly and and beat Cao Cao as fast as possible when you are ready to leave. Just remember to talk to the Mystic after each run. Silver chests (covered in a beam) require a sufficient amount of Miasma in its area to become available and tend to contain more rare rewards. Also some are Hidden until you activate a related Dragon Portal near it.)
(This will also clear another Achievement.)
-Easily get all Mystic Weapons (+1 to +81): Gauntlet Mode > DLC Stages > Heavenly Emperor's Palace (Hundun drops a Mystic Weapon for EVERY member of the active team. The longer you fight him each time, the better the weapon quality is, ranging from +1 to a max of +81 and often having several rare Orbs. Take in someone with a functionally infinite setup of Samurai-style Ultimate and safely wreck him along with a new set of four team members each time to get everyone their ultimate weapons. You can fight him immediately after passing the first Crystal, you don't actually have to do all four. They just lower his Max HP and slow the growth of Miasma that can full-heal him a limited number of times.)
(This will also clear another Achievement, FAR more easily and quickly with far less frustration than the normal way.)
Note: Using the Solstice Formation gives the Team Leader the Getaway power. This teleports the team directly to the Escape Point (except on a very few restricted stages). This may take a bit to unlock, but saves a massive amount of travel time when you are ready to finish a stage.
Activate a character's current Formation Power by hitting [Press in Right-thumbstick]. None of this is explained anywhere that I recall and that is stupid since they are very useful powers. It drains an amount of the Team Guage, which is limited by the number of active characters and recharges both passively over time and while using Dragon Portals.