How do I repair a door frame?
Assuming that the problems you're trying to solve are 1) stripped hinge screws, and 2) splintered jamb veneer...


You can mount the hinges in one of several ways: 


Fill the holes firmly with wood slivers (or toothpicks) and wood glue. Allow to dry, and drill for standard hinge screws (commonly #9 or #10 by 1" flat-head). Use a pilot bit that's slightly smaller than the screw shank (not the thread diameter).
Purchase repair screws with a larger thread diameter.
Use longer screws that reach to the framing. 3" gold construction screws often work well and have a similar appearance. Be sure to not drive them so tight that you move the jamb. You might want to pilot with a 1/8" bit to prevent veering or splitting of the stud as you'll be fairly close to the edge with the outermost screws.

Since your jambs are painted you can use any common wood filler to skim the damage. Use a tool or nail to simulate wood grain with light scratches, them prime and paint.