A few thing you should know about SEO.
First, good SEO can't be achieved with a one time fix. Good SEO is something that is achieved overtime. Sure, someone could get you to that first page on Goolge, but if you maintain your SEO you'll loose that spot quick.
The next thing is content. The content of your site has to reflect the keyword searches, and the more specific the better. If your content doesn't reflect the keyword you want. Then your rankings will go down.
Now on to link building. You have to be careful with this. There's a fine line between link building and spam. Suddenly flooding forums, facebook and twiiter with "Come check out my site" links is spam. Not only will your rankings be penalized for that Google may ban your site all together. Now if you add a new game to your site and tweet about it and 100 of your followers re-tweet it. That's link building.
The best way to get your rankings up is to have a popular site. I know that sounds counter intuitive, but that's how it is. The more people visit your site the higher your rank will be.
With your site. Well good luck. Online games might be one of the most popular type of sites on the web besides porn.
If your site was made correctly you should already have all the SEO on your site that you need. If it's lacking in SEO. You should hire a credible web designer or design firm to fix it up.
And to be honest your best bet to getting more traffic is to get into some type of click-through ad campaign. Pay the money to get your ads on game related sites, and don't worry about google ranking. The internet market is so full of online game sites that there's a good chance that you'll never rank on a first page search. That is unless you have something truly totally unique.
One last note, be careful of who you hire. There are allot of scammers out there that will use techniques that can get you banned from google searches.