Neither
The correct form is :
Alex always does his homework late at night.
Here the simple present is used to describe a habitual action - that's not the only use of the foorm, but it's probably the most frequent one. If you just want to describe a habitual situaion, this is the form you should use.
The continuous form can be used to describe habitual actions but it's nearly always used to express annoyance or to suggest there's a problem :
I don't want David here any more. He's always causing trouble.
I never see her because she's always working.
In your example, a possible context might be :
I can never get to the computer late at night because Alex is always using it. He leaves everything to the last minute so at midnight he's always doing his homework for the next day. It really annoys me.