Lawfully Good would think that this city to have some defence against these immortal bombers, so she would fight against them in hopes of finding their source while defending the city. The reward of a job well done would be enough of a reward for this character.
Neutrally Good would attempt to find out how to join the bombers, and do so, reporting back to the employer as soon as possible without causing a scene. This character wouldn't ask for anything in return, but want as much information on the bombers as possible.
Chaotic Good would just join the bombers, destroy their base, and report back to the employer. Hoping to kill off as many of the bombers as possible. Maybe they're only immortal when they're bombing? This character would be enticed by the immortality.
Lawfully Neutral would only intervene if the bombing broke any laws. This character would most likely be too busy otherwise with helping this town get itself back into a lawful state anyway.
True Neutral would really just take the job to get the immortality. Once that has happened, she would probably leave town with her new found immortality.
Chaotic Neutral would ask the employer for money (up front) and join the bombers, siding with whichever one was most fun. Once invincible, probably going to join the bombings until bored. Once bored, this character would blow up the bombers' HQ and leave.
Lawfully Evil would organize the crime, ask the employer for money, talk to the bombers with threats of jail or blocking them in their HQ. Once quelled, and money gotten, this character would then use the bombers to extort "protection money" from the townsfolk.
Neutrally Evil would screw everyone over. Join a crime ring, kill the employer, join the bombers to get that amazing immortality.
Chaotic Evil would be going into the town whilst trying to kill everyone in there. If the employer somehow evaded the killing spree, the interview would go horribly, with the employer dead after telling this character about the bombers. Immortality gained, and killing everything with it.