"그는 자면서 말합니다."
Translation:He speaks while sleeping.
그는 is always he, 그 can be he, (to) him, or his according to the context. He can be 그가, but I think 그가 is also used for him like 그를. Still, be consistent. 그는 goes with 그녀는 . . .
These words are determiners (well, except perhaps his/hers?, but they might as well be in a reflexive way). The sense of the particle is included as I see it.