This is rather straightforward, just follow the Main Story markers.
There are four different ending battles with two different endings, depending on if you do the final battles in the base game or the Final Horizon update.
1) Clear the game on Easy/Normal
2) Clear the game on Hard/Extreme*
3) Clear the Another Story (from The Final Horizon update) on Easy/Normal
4) Clear the Another Story (from The Final Horizon update) on Hard/Extreme*
Note 1: I don't think the Final Horizon ending will actually trigger the achievement, but I'm including the details just in case it does or is helpful to somebody.
If you are doing the Final Horizon battle, there are two things that make that fight particularly frustrating (because they aren't hinted as an option, let alone being a required action):
1) You have to directly target the cable on The End's head to disconnect it and prevent its auto-healing... which can be quite difficult with all the attacks constantly trying to knock you away and end your combo. A way to lock on with decent accuracy seems to be to lock onto its head then start a combo and dodge to either side after the first hit. If you are lucky, it will switch targets to the cable and (if you manage to destroy it before an energy ball sends you flying) you can finally make progress in damaging The End.
2) After you get The End knocked to the ground, you have a short period to... do something. (I never noticed a hint and had to look it up. I don't count the cutscene where Super Sonic does it with the Knight's sword as a "hint".) You have to Quick Cyloop (I don't think you can do a normal Cyloop) on the targetable spot behind The End's head, where its giant gun is stored. This will pull its gun out to fall to the ground. Then you have to do another Cyloop around the gun (normal or Quick Cyloop, both work) to enter Phase 2 of the battle.
Note 2: The difficulty only matters when entering the final battle(s). You can change the difficulty (up or down) at any time prior with no penalty.
*Extreme is unlocked by earning an S-Rank in all of the Battle Rush modes