Go with a sexual motive (Which is why serial killers kill (not counting people with psychosis) , there's always a sexual element, don't mix up spree killers and serial killers.)
The woman could be a vulnerable victim e.g a prostitute, or someone who relies on care. And she could fit his type, example being having long brown hair that's parted down the middle.
As for why he kills, he could be a necrophiliac. And enjoys the power he has over them, an example of this could be dressing the body up after death. Or it you want someone who's sadistic, then that would be all about torture.
To say someone kills because they're a psychopath would be wrong.