Do i have to play all parts of the acts like 1-1 1-2 1-3 1-4 or can I join in 1-4 beat the act and go to the 2nd?

You have to beat EVERY chapter of every act in the game as Dom to get the "I Can't Quit You Dom" achievement. They don't necessarily have to be in order. For example, you could beat 1-5, 1-6, and 1-7, then go back and do 1-1, 1-2, and 1-3. Given this scenario, you'll get the achievement at the end of 1-3.

This is essentially the same as the "beat this act on a difficulty level" achievements. You can't just go to the last chapter of an act, beat it and have it count. Epic anticipated this.