Can I run a ground wire from a cheater plug to another 3-prong outlet to ground it?
The problem you may encounter is that in many areas, even with a homeowners permit, you have to rip drywall to get grounds to the ungrounded outlets to properly fix them.  It sounds like the ground makes it to the first outlet in the chain, so you may wish to consider replacing that outlet with a GFCI and putting the other outlets on the load terminals.  This alleviates most of the safety concerns of the ungrounded portion of the circuit.  GFCI breakers can be used as well although they are often more expensive.
I'll leave to someone else to figure out the legality of stringing a surface ground, but my instinct tells me done properly it would be better than nothing.  It would be wise to add information about where you live for a properly informed answer.