The Marine (Corporal) is an unnamed playable character in Doom II (he may have been the inspiration for Stan Blazkowicz of Doom II RPG).
Background
Back in the 21st century, he was trained by Sergeant Workman and his C.O. was Sergeant Swann.
He was the only survivor of the invasion of Mars Base which took place concurrently or shortly after the invasions of Phobos and Deimos.
He was back on Earth at last, after days of hard fighting in space. He had returned home on well-earned leave. He was one of Earth's crack soldiers, hard-bitten, tough, and heavily-armed. When the alien invasion struck Mars, he was the first on the scene. By killing, killing, and killing, he won. He stopped the invasion, saved Mars base, and became a war hero.
What they don't talk about so much is that he was the only survivor. But that's all behind him now. He quit the military, and was heading home. His drop pod landed with a crunch. He opened, and looked out. He found that the city ahead of him was on fire. What the devil is going on? He staggered forward, clutching at his sidearm. Packs of refugees were fleeing the flaming metropolis. A band of them shriek in terror. He squinted. What's that? Someone was attacking the refugees. He rushed up and blasted away, killing the troublemaker. It looked like a human, but something's was wrong. His mouth was filled with half-chewed flesh, and he was all messed up, like a zombie from a bad horror movie. Hell, not again! The Marine could feel it. It had all starting again, just like on Mars. First, people were taken over, turned into cannibal Things. Then the real horror starts, the deformed monstrosities from Outside. But now it was on Earth.
He was quickly recommisioned back into the military, by his earth commanding officer Sargeant Swann, who issued him missions objectives to save as many people of Earth as possible, and to stop the invasion.
Behind the scenes
Depending on the materials, the Marine of Doom II is a seperate character than the original Marine in Doom/Thy Flesh Consumed. Although some official sources, and later manuals often conflate the two characters (or even later characters in Doom 3/Resurrection of Evil) all into one character.
This article goes with the original backgroudn story covered in the Doom II manual, and expanded on in the Doom II: The Official Strategy Guide.
Also worth nothing that he is a 'Corporal' and his commanders are Sergeants. Where as original Doom Guy was a Sergeant.