As a gamer of more than 20 years, your behavior is abhorrent, and a shining example of what types of behaviors toxic people/players exhibit; the type that should be purged from the community.
No, you should not bring this to HR. You should not bring this to anybody.
You didn't mention whether or not the name or logo of the company was displayed during the stream, so I feel safe in my assumption that it wasn't. At that point, HR probably doesn't care. Telling them any of this will only direct the spotlight on you. A senior colleague, who said some things that could cause offense while being harassed, is probably a lot more forgivable than a new employee, just out of college, going out of their way to harass a senior colleague, that they just met.
Let me try to pick this apart.
As I am searching for streams to view, I end up finding the most senior member of my team streaming.
This could have been a great source of camaraderie between you too. But you blew it. I don't believe that you can even mention that you play the game now, without slipping up and telling him something about the match you were harassing him in, which will make it very easy to identify you as the harasser.
Of course, being the idiot I am, track his game down and join it. I then proceed to single him out and destroy him in the game since I am a lot better than he is.
No, you weren't being an idiot. Idiots don't know what they're doing. You were being a troll and trolls know exactly what they're doing.
You singled out your coworker, in a game that he probably enjoyed playing, and harassed him. I've said harass a few times now, some people might think it's being too harsh, or using the wrong word. I'll just refer to this bit here
I then proceed to single him out and destroy him in the game since I am a lot better than he is.
Each time, he gets more visibly mad, which causes me to continue...
So you've been watching and harassing this guy for long enough to see that your actions are making him visibly mad; and that causes you to continue harassing him? Why would you continue a pattern of behavior, that is making your senior colleague visibly mad?
The only answer I can consider to be true, is that it was your intention all along to see how far you could push him.
...until he starts a very specific sexist and racial rant against me based on my username. What I thought was harmless fun ended up showing me his true colors.
Not that I particularly care for the answer, but how could you consider anything that you did to be harmless fun? At each and every step, you have done the wrong thing.
You've harassed this guy, to the point that he probably (fairly, I might add) felt that he was being personally attacked, and now you're "offended" when he personally attacked you? It "offended" you so much, that now you're trying to punish him for his comments by bringing it to HR?
I found his behavior extremely offensive
I don't know if I'd be able to find his behavior more offensive than yours. Especially not "extremely".
When people are being harassed and their heart rate is high, and they're stressed, and they've got that fight-or-flight mechanism kicking in, and the harassment does not stop, they'll probably say things they don't mean.
They'll probably think that all of the other tools at their disposal have failed (because they have) but the situation is not getting better. The antagonist hasn't stopped. What other things are left that could stop the antagonist? When you're under that pressure "offend the antagonist" probably doesn't sound too bad.
I only hope that your coworker finds enjoyment in the game still.