As opposed to the four starting factions, to win with the Tomb Kings you need to gather five of the nine books of Nagash and then fight a final battle at his Black pyramid. Honestly, that makes this campaign a lot easier, since you wont have to do rituals and are not competing with the other main factions for the same ressource.
For you to note is, that the books are either with rogue armies or in specific settlements. That makes diplomacy, especially on the higher difficulties, an important thing to use correctly.
For the rogue armies you will want to have a Military access treaty with the faction(s) in whose regions the armies roam around, otherwise you might start unnecessary wars and have to fight more than you can chew. The armies themselves are a bit strong for the beginning, but midto-endgame armies should easily beat them. Rushing them with two armies will make them easy as well.
The settlements on the other hand requires the opposite diplomatic approach: You don't want to be in an alliance or non-aggression agreement with the holder, since you need to take their settlement for the book.
If you have multiple books you can reach equally easily, you should look at the advantages the impart on you. Some are (only) useful early on, some are good for your heroes, others for you economy. Something to keep in mind.
After that you have a final battle against multiple undead armies (with two allied ones supporting you), which isn't to hard if you prepare properly. Since they are mostly Tomb Kings you will have to take on a lot of light infantry and well armoured monsters. Use Ushabtis and Tomb Scorpions against the former and Bone Giants against the latter.
To remember with Khemri is that they require little money for armies, but a lot for settlements. They also require buildings to determine how many of a unit they can field.
That has three major impacts compared to other factions:
1. Whenever possible get money. Sack, Loot, go after seatreasure, take whatever you can. You will need it, especially early on. Later on however, in the endgame, you might not need it at all, so you can easy of on it.
2. Do not let your provinces be razed. You will need a lot of territoy to field multiple high level armies, and building up the necessary buildings is costly and time consuming. And money is not that easy for you to get.
3. On the opposite side, if you concentrate on the corresponding technology for the amount of Lords you can have, you can field a lot of armies quickly and just rush many of your opponents. Since you don't pay upkeep, it works fine that way.
I think Settra is the easiest one to win with, as he himself is very strong and starts in a good position.
Arkham might be quicker, since he already has a book, but diplomatically he is in a bad position.
The other two are harder to do, since they start in positions that are a lot more contested.