The hero in "Gilgamesh" was a great hunter who wanted to kill a creature that was servant to the gods. If I remember correctly he was trying to kill this creature for honor, but that's only from my assumptions and inferences. Well a modern day video game character to which i find similar is the man named Wanderer in "Shadow of the Colossus". A man rides out into forbidden lands to save the one he loves. He must kill giant ancient creatures to bring her back. That's the only character I can think of that is similar. Here's a link to the official site. http://www.us.playstation.com/PS2/Games/Shadow_of_the_Colossus/OGS/
Now the story "The Illiad" is about Achilles, a powerful man that is invincible to everything except getting hit in the ankle. They actually made a flash game of him on addicting games.com, I'd say visit it but you can't while in school because it is blocked. It's a game where you just use your shield and spear to decimate your enemies armies for honor and glory.
http://addictinggames.com
Now here's the thing. I've compared each story to a game. I have found that modern day characters, this is my opinion not every one's, are a lot alike with the old stories. There a lot of games based on old lore's. "God of War" for one, I don't think there was actually a guy named Kratos, but it does have a lot of the Greek mythology and how they acted. So I am thinking they are based on the old stories.