I have been riding bikes and motorcycles since I can remember. Doing no hands is no problem, my question is with this particular generic bike frame that I built: I cannot do no-hands riding. Is the frame straight, or is the problem with the fork? I have another road bike and a mountain bike.
It may not necessarily be the untrueness of the frameset that's preventing you from riding hands-free. As was already said in the comments, it's the overall frame geometry that determines that, and in fact it's one most important parameter.
I've found this very informative picture at http://cog-nitivedissonance.blogspot.com/2012/01/low-trail.html:
It is the trail that determines how easy it is to ride hands-free - the longer the trail the easier it is. The trail in turn is determined by the head tube angle and the fork rake (fork offset).
As I said earlier, the longer the track the more stable the steering of the bike is. This characteristics was used in e.g. motor-paced bicycle speed record setting. Have a look at the picture of Alfred Letourner on his record bike (found at http://piedmontvelosports.blogspot.com/2011/08/alfred-letourner-set-motor-paced-speed.html):
Notice that the front fork seems to be reversed providing a negative rake that increases the track even more.
I'd be suspicious that it's the wheels more than the frame that make your bike unstable. First confirm that the wheels are true up/down and left/right with the tire off. Also, wheels that aren't properly dished won't be centered between the forks and brakes, making your weight unbalanced. Dishing can be corrected by tightening/loosening all the spokes equally on one side, but it's difficult to get it right - better left to an experienced professional with the right tools.