There will most likely be 2 different achievement lists, 1 for X360, and 1 for X1. Campaign will work the same way. If you have a game for both consoles, you can still use the same account to play them both, but consider the 2 games not compatable with each other.
Example: You beat the campaign on the X360 and get around 500 gamerscore. On the X1 you just play multiplayer and have 100 gamerscore. You will 600 of the possible 2000, 1000 for each game. Multiplayer, singleplayer, etc. will be seperated into 2 different games.