T. Washington was a Marine stationed at Mars City.
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, though he was most likely killed or turned into a zombie during the demon invasion.