There are many ways in which to break into the Games Industry and trying a mixture of all will help with your success.
To get a gaming job you need a solid strategy and a clear plan on how you aim to achieve your aims.
There are some great websites and forums that deal with the Games Industry. As an example, if you work within Games Art then Artstation is the place to be.
On Artstation you can create your own fantastic portfolio with it's own unique URL. This saves messing about trying to create your own website, which may not do what you want it to do.
Why not work with some old college mates and build your own Indie Game. You won't necessarily get paid for your work, but it will help you to add experience to your portfolio and show that you are serious about landing a games job.
GameJams are the ideal way to showcase your skills to game studios. They are game building competitions where you can show off your skills and create quick, addictive games.
Many people in the games industry landed their first gig testing games. It is a more junior starting position than a developer, but is a starting point and something to add to your resumé.
Landing this games job will enable you to build up a network of valuable colleagues and showcasing your skills in this could help you make the move across to your ideal job.
Many game studios have internship positions and although it can often be very competitive to land a position, if you do it gives you some much needed experience within a top class studio.
Not every studio will have an internship program but most are open to the idea of a part-time, temporary, inexpensive (or free) worker that could help them finish their game