Should Alistar be getting kills?

Solution 1:

Don't worry too much about trying to get kills - Alistar's role in a match is to provide support and disruption. You may not get many killing blows, but if you're a decent player, you should have many many assists instead.

If you're worried about losing out on the money that comes with kills, I'd suggest you look into Gold/10 items on Alistar - Philosopher's Stone and Heart of Gold being the better of the 4 for him. (Duplicate items don't stack, but there's nothing stopping you from picking up one of each).

That said, Alistar makes good use of Sheen, which is a cheap way to double your basic attack damage, and gives him mana to boot.

Solution 2:

If you're playing as a support/tank, then kills are not your goal. In addition to gold generating items, I'd also suggest the masteries that boost gold or exp on an assist.

Before the V1.0.0.116 patch, it wasn't uncommon to see AP builds on Alistar. His ratios have been reduced pretty significantly, but it's still viable to build AP on him to increase his damage. Try it in an AI game first though. =D