You will find Diablo on the last part of the fourth Act. (ACT IV)
Killing Diablo is really straight forward, though it all depends on your items, difficulty you are playing and the level of your character.
Diablo has 3 phases in order to kill him, the first is to bring his health to half and then he will transfer you to the shadow world then for the second part, you will have to deplete his health and also fight your shadow clones (1-2 times). Then you will return to high heavens and you will have to defeat him once more completely. Once down you will unlock the achievement.
If you are going mainly for the achievements, then i would recommend for you to play the whole game on Normal in order to get the achievements easy and fast.
If you have played the Vanilla version of Diablo3 then it would be even easier as your character would be transferred and it can be used for the new campaign. Even though i choosed to play on master so i could level up faster. That is up to you and your skill/experience of the game.
If at any point you find Diablo too hard, then you can always lower the difficulty of the game, while in game and make things easier.