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.

"The battery of Paul's laptop doesn't work well" is correct. But "Paul's laptop's battery doesn't work well" is also correct - you just don't need 'the' (definite article) when you have a possessive.

I agree with @topo morto and Phil. 1. "The Paul's laptop's battery" is wrong for several reasons. You can't put a 'The' before a person's name, and you wouldn't use the " 's" form with an inanimate object such as a laptop. 2. "The battery of Paul's laptop" is fine. 3. "Paul's laptop battery" is the best phrasing because it's the neatest. The object is a 'battery'. 'Laptop battery' tells you what kind of battery it is. 'Paul's laptop battery' tells you what kind of battery it is and who it belongs to.

the first (also: personal proper names {Paul} do not take an article)

They are all right.