Dumbledore was not only a close friend of the Potters; he was also the one who recommended the Fidelius as their best chance. It must have been a rude shock, when their best friend betrayed them. Dumbledore is also portrayed to be a skilled Legilimens, when he uses legilimency on Kreacher in Harry Potter and the Order of the Phoenix.
“And Kreacher told you all this… and laughed?” he croaked. “He did not wish to tell me,” said Dumbledore. “But I am a sufficiently accomplished Legilimens myself to know when I am being lied to and I - persuaded him - to tell me the full story, before I left for the Department of Mysteries.”
Dumbledore was also Chief Warlock of the Wizengamot. He had used legilimency on Morfin Gaunt, as he had access to Azkaban inmates, owing to his position in wizarding society.
“Yes, but it took a great deal of skilled Legilimency to coax it out of him,”
Why didn't he try the same on Sirius? Why didn't he use Legilimency on him to check whether he was actually guilty?