According to this fan-review from Aintitcool news, Whedon addresses this issue at a comic-con Q&A session. In short, in 'Buffy' Angel gets his money by stealing it. He might be a hero, but he's evidently not above simply robbing people when he needs cash.
If it helps, he probably justifies it by pointing at the good works that he does.
An audience member asked where Angel got all his money. Whedon joked,
“He robs. We’re not going to SHOW that.” When the same person asked
where Angel gets his clothes, Whedon shot back, “Clearly at one shop!”
For part of the first season of 'Angel', our eponymous hero seems to receive some funding from Whistler (note that he was homeless when he met him) and in later episodes he runs a detective investigation agency with Cordelia in charge of debt collection as well as recovering some cash that he left stashed in the Hyperion hotel in the 1950s.
After they move to Wolfram and Hart, he seems to draw a sizeable salary as Managing Director of the local Division.