I am adding a new eletrical outlet in my finished basement. The good news is the wall I am adding the outlet to has a closet behind it so I can get in behind the wall to run the Romex, mount the box to a stud, cut the hole for the box, etc. My question is, the closet has a light in it can I tie in the electricity for the new outlet to the existing electrical going to the light in the closet or do I absolutely have to run the wiring through the ceiling and wall to tie it into an already existing outlet?

If the light has constant power in the outlet box, then fine extend form the light. Be sure you don't connect to the light, assuming the light is controlled by a switch. If that happens, then the new receptacle will only work when the light is on.

How do I tell if the light has constant power?