Not really. Umbrella has been shut down in the game universe. (which is the game the executioner appears in).
He's a result of a parasite called Las plagas (variation of it) which originated in Resident Evil 4. They don't give a specific origin but there is a pretty run down butcher shop amongst the village with a lot of carved up corpses and a couple tools like the one he wields (Which is basically a giant meat tenderizer combined with an axe) hanging on the walls, leaving you too assume that he was a butcher at some point. Why he wears the mask and puts nails through his head is anyone's guess.
I like to think he was trying to get the parasite out of his body before he became too infected, so he started driving nails into himself. Or he was just a sadist to begin with...so he became a butcher to satiate that need. They leave it open for interpretation mostly. Which makes him so much more interesting and creepy IMO.
For the movie I just think a long the same lines. Might be a little harder to believe someone like him is living in LA though...but who knows