T. Washington was a Marine stationed at Mars City (2145).
When the player meets T. Washington in Doom 3, he greets you and offers some advice about your stay on Mars. He is one of the friendlier Marines the player encounters before the demon invasion. After Sergeant Kelly gives you your first assignment, you can find Washington reading a computer terminal in the lobby.
His fate is unknown.
It appears that T. Washington actually survived, at least until the later invasions on Earth (2163). He was the writer of the report discussing the analysis of the Super Gore Nest (found in the The Story Of Earth codex).