You need to set some reasonable goals. "be able to play video games" is too broad - that covers everything from being able to figure out which button to mash to dealing with text-heavy games with five screens fully of back-story written in Ye Old Faux Samurai.
In three months you're not going to make a huge amount of headway in terms of what you need for anything very text heavy. Games and other forms of fiction can have a wide range of grammar including unusual/non-standard language (me caveman me talk funny, Ye Old Immortal Quest Giver-speak, etc), unusual vocab you don't find in other sources, and so on.
You can focus on picking up the major vocab (basically operational stuff like 経験値 or whatever) for some of your games, provided you do this as a sideline to other study. Use a 攻略 (walkthrough) website as another source of reading material. But any plan which involves just trying to translate as you play through, while you're still at a relatively low level, will probably just end in frustration.
The main point about self-study is that you need to be able to set yourself tasks and go through with them. For Japanese (or any language study), you need to be studying daily even though nobody will tell you off if you don't and nobody's setting you any homework.