I'm assuming Mafia Wars is pretty much the same as Mob Wars, which I play.
The way hitlists work for Mob Wars, someone puts them on a hitlist, and you have to be in that section and attack them from there. Attacking them regularly, even when they are hitlisted, will not allow you to kill them to get the bounty.
Though to be perfectly honest, hitlisting is tough to do unless you have a LOT of members with high powered weapons/armor. But you don't have to win the fight to kill them, it just makes it easier.
Also, using IE is better, since you can hit the attack button multiple times.