I'm not sure if this would fall under the classification of "biggest video game achievements" but this is the only thing for me that comes close: I started playing Counter Strike as a young teen and first broke into the game back in CS 1.3 way back in '01. Just casual games and public servers. Did that for a couple years until I found out what competitive CS was. Slowly got into that, playing with friends at first, then started taking it more serious as time went on. I couldn't play real sports anymore due to injuries, so this was the next best thing, since I'm a real competitive guy.
After a few years I slowly started to rise through the ranks of the league to where finally, at the end of CS 1.6 and the beginning of CSGO, I made it to the professional level. By this time, I was all over the country attending LAN tournaments, getting free computer gear from sponsors, and receiving a small monthly salary. Then I got old, my skill diminished, and real life took over, so I hung it up.
So I guess, yes, this was my biggest achievement in video games. From around the age of 14/15, up until 32, I played only one game, dedicating myself to become better and trying to reach the top. And I can say I did. To play with and against some of the biggest names in the game, is something I'll never forget. I looked up to these guys when I first started playing. Mimic'd my game after them. Tried to teach myself how to play by watching them. Fast forward 10+ years and these guys I once looked up to, are now some of my closest friends. What a ride.
/endrant