I know some people read the title of this card and went bananas. But let's be honest, not all men play video games. Same for women as well. In my own personal relationship, I am the ulimate gamer and my partner is a step or two down from me. We play things like Left 4 Dead 2 or Gears of War 3 together. I myself jump into other areas of gaming, such as resident evil, Saints Row etc. I fought this "How to" add online and I thought it was funny.. Read through.. And here is your question.
WHO IS THE GAMER IN YOUR RELATIONSHIP!
Find out what types of games your boyfriend likes to play. Does he like role playing, racing, sports, mind, or sports games?
Find out why he doesn't play games currently. Does he just not have time and now needs a push from you? Does he have an emotional attachment that he does not want to recall? Is he afraid of losing? Does he have an injury that prevents him?
Work with him to deal with the issues from the aforementioned step.
Convince him that video games are the best way to spend time together without wasting money. Also invite him to play at your house so he can get to know the way you live and the way your family is. If they don't know about him, this would be a great introduction as a Friend. Just remember he might be nervous at your house with your family.
Find a few games he likes and play the ones he is better at. Mix it up, but always go back to the game he is good at. Racing games are generally easier for you to lose at without showing it.
Be supportive of him as he plays. Image titled Get Your Boyfriend to Play Video Games Explain to him how much fun you are having with him.
Step 7 Have fun !
Personally, I have dated guys that did not like video games, sports and comics. He was more into dancing then anything else. The old saying you can lead a horse to water but you can't make them drink applies to the situation I had with that fellow. He showed me the dancing world and I tired my best but when he crossed into my world (Hobbie stuff) it just was not him. So we ended up not taking our relationship to the level of exclusive and we remained friends because one thing we could do is make each other laugh.