To execute a FADC (focus attack dash cancel) you first need to focus attack out of a move. Please look at this answer which moves can be canceled.
Bascially you execute a move and after you hit and you are still on the ground you press MP and MK to cancel the move. Try it first with Hadoken, as this is usually easier to execute. So just train Hadoken...FA. If you succeed you will glow yellow and you will cancel the animation of your previous move and make a focus attack.
Once you feel comfortable with the timing, then hold the FA after the Hadoken and press forward twice, as if you want to dash. Then release MP+MK. You can of course also dash backwards.
Go in training mode and display your input. If you tend to press something like MP before MP+MK then you will not FADC. Try to vary the timing. Usually you have to FA after the hit, but it is not enough time to "hit confirm", just assume that you will hit and use the timing from your muscle memory.
In the end it just boils down to training and crisp execution.
FYI: You cannot FADC Ryu's HP Shoryuken.