In the scene where Clemenza and Tessio ask to split off from the Corleones and form their own family, whose portrait can be seen hanging on the wall behind Vito?
And why did the director choose the portrait of this man?
Although, it is highly unclear who's portrait it is, it resembles to one of Rembrandt Harmenszoon van Rijn's (a Dutch painter and etcher from the 17th century) self portraits. It can also be a rework by someone based on Rembrandt's portraits, but that is unclear as well. By the image it looks like it "could" be a self portrait of Rembrandt as he perceived himself as a young boy, or by someone (unknown) who perceived this as Rembrandt's younger self based on his self portraits.
Although since it's also unclear why the director might have chosen this painting but one of the reason would be to represent a royal (although evil) presence of the Corleones, an influential presence in the Golden Age of Mafia establishments. The fictitious Corleone Family is also based on a real life Borgia Family who were primarily involved in politics and prominent during the Renaissance in Italy. Since, Italian Renaissance art and Dutch Golden Age art have conquered their prime positions in the art history books the representation of a Dutch Art piece (possibly mere fake or just a re-production) by a Great Dutch painter connects it to that era which influenced or inspired Mario Puzo while writing of the fiction. Although, this is just my perception, but a rather possible one.