The resume is most likely your first real point of contact with your future game development employers. As first impressions are vitally important, a well designed and impressive resume is absolutely crucial to one day being able to make professional video games. There are a couple of insider tips to remember when applying for a games job; ignore them at your own risk.
When writing and editing your game resume, convince yourself that you deserve the job. You are a talented, creative, hard-working individual who is dead-set on getting into the games industry one way or another.
Do not be intimidated by their job requirements. Almost every job listing you're going to see will "require" 3-5 years of experience. That barrier to entry is why it's called breaking into the industry in the first place. They may ask that you have years of expertise with dozens of programming languages, shipped numerous titles, blah blah blah. Those are the characteristics of their ideal candidate, not necessarily the applicable they're going to end up hiring. Do not be the one to say no to yourself; make them tell you no.
You also want to show that you have done the things that will occur as you actually make professional video games. These are activities such as working with others and a team, using large code bases, receiving feedback, and working under deadlines. All of these are things that will bring attention to your resume, because they will make it seem like you've already been working in the industry! And when you're competing against people who actually do have some work in the industry, that is a very good perception for your future employer to have of you.
In addition to a resume, most companies also accept a cover letter that expands on how you want to make professional video games. This is your chance to sell yourself, telling the recruiter what you'd like her to know that does not fit or is not appropriate on the resume. For example, you can name some of your best personal qualities and characteristics that make you an especially good candidate for the job you want. You can also summarize your experience in a way that demonstrates why you're an ideal candidate. Ask yourself: what do I want them to know about myself that does not go in the resume?
Now that your resume is ready, how do you find companies to apply to? There are literally hundreds of game development companies for you to make professional video games with. The best strategy is to prepare yourself for the long haul. When contacting game companies, be persistent; contacting companies and applying for jobs takes about as much effort as developing a game in itself! You might get lucky and get a hit on the first company you apply to, but this is the exception, even for incredibly talented applicants. You might have to contact twenty companies before you get any kind of response, and you might have to interview with four or five before you get an offer. Do not give up!