He is very dutiful and protective of the Jarl, both as her Steward and Husband, he is concerned with people bothering the Jarl with their petty concerns and states that his only concern is keeping his wife safe.
Aslfur can give a bounty quest if asked for work and can also give rewards for completed bounty quests in the hold.
If you kill Gorm, Aslfur will send you a letter thanking the player for doing it. After receiving the note the player can speak to Aslfur and mention the note. Aslfur will acknowledge he sent the note, and tell the player this conversation never happened.
The Elder Scrolls V: Skyrim (First appearance)