When he first met the Doctor, Frobisher was a private investigator under the alias of Avan Tarklu. A huge bounty was placed on the Doctor's head by the ruthless capitalist, Josiah W. Dogbolter, but instead of handing him in for the reward he decided he liked Time Lord and helped him steal the money from Dogbolters estate. He then joined the Doctor for his journeys through space and time.
It was soon into his adventures that he decided on the name Frobisher because it sounded British and hoped it would please the Doctor.
Frobisher appears in the IDW comic book series 'Doctor Who: Prisoners Of Time'. While the Doctor that contacted Frobisher for help was not one he was familiar with, Frobisher was familiar with the idea that what someone looks like on the outside does not alter who they are on the inside.