Should I paint the figure parts first or build the figure the paint?

That really depends on you and the figure. I usually start by painting the head. I keep it separate until it is all done. I then attach legs and torso together and pain that. Then the head gets glued into place. As far as arms go, if the figure is meant to hold a weapon or something I usually will paint them separate and glue them to the figure. Otherwise I will glue them on at the same time as the torso and legs and paint everything together.
Equipment and weapons are painted separately and glued on after the fact.
That's just how I do it.

As with Marty, it's head first, separate if possible. The rest of the figure depends on position (can I get proper highlights and shadows assembled) and engineering of the figure. If dryfitting shows I'll be able to paint before assembly, then I might do that more readily.