5m down and 3m up is more than enough to run a online game. Games use very little bandwidth. Most are well under 1mbps to allow people that have only crappy DSL as a option to also pay money to the company to play the games.
Your problem is likely you are getting interference in addition to poor signal strength. This interference causes the latency/ping to not be consistent. This is what causes the lag and other issues in games not the network speed.
There is not much you can do. You can try to change the radio channel on the router but that seldom helps any more. Maybe turning you pc so the antenna more directly face the router or using something like a USB extension cable or antenna extensions might help...I would not spend a lot of effort because it is unlikely to help.
In your case I would consider powerline network devices. Things based on the av2 standard work in much better in most houses than the older av200 and av500 technology. You will likely get much more bandwidth but more important for games you will get much more consistent latency because powerline as susceptible to outside interference from neighbors signals.