The run-out could be in the quill, the chuck, or the arbor. First, remove the chuck (I'm assuming that it's mounted on a morse-taper arbor) from the quill using a drill drift, and run the point of your dial indicator on the inside of the quill - on the wall of the morse-taper socket. If the run-out is there then the drill is junk for sure. If the socket is good then the problem is in the arbor (cheap to replace) or the chuck (not cheap to replace, but often worth replacing just to have one that works smoothly). It's also possible that the chuck might not have been assembled properly onto the arbor, or the arbor might not be seated properly in the quill. If you can remove the arbor from the chuck then you can install it in the quill and check the Jacobs-taper surface for runout. The arbor is, honestly, not likely to be that far out, even a cheap one. Some chucks have a hole in them so you can bang the arbor out using a pin punch. If your chuck doesn't have that hole then you either need to drill such a hole (often requiring another drill press) or you need wedges made for the purpose to remove it.

Once you know where the problem is you can work on it.

As to how much run-out is acceptable, I'd say that 2 thou total runout is reasonable for a cheap drill press, but 10 thou is actually well within the needs of the average woodworker just drilling holes in wood. However! Just because you don't really NEED the accuracy doesn't mean you should settle for junk.