The most natural phrase would be "Paul's laptop battery doesn't work well."
Explanation: The battery belongs to Paul. (So does the laptop, but that's linguistically irrelevant here) The noun "laptop" is used here as if it were an adjective. to qualify / modify the noun battery -- you could also say "the battery of the laptop," but it would sound overly formal. We don't usually say "the laptop's battery" because the laptop is inanimate, and isn't generally considered to have possessions -- both the laptop and the battery belong to Paul.