1 - Build a portfolio
The game development industry values doers and makers. Show a potential company that you can make shit happen by creating and maintaining a portfolio of related projects that you've worked on. This includes games, code, design, art. A good portfolio should show progress as you mature in your skills and have diversity in the projects you've worked on.
2 - Pick a niche
Developers are relatively easy to get hold of. Developers with a physics background who can work on physic engine code? Not so much. Pick a related topic that you can become an expert at and you'll never have a problem finding a job. The harder and more math in the topic, the better. Good ideas are: Phyiscs, rendering, data science, high efficiency networking. Generally, as a developer in games, following the math is a pretty good advice.
3 - Educate yourself widely
You can write ultra-fast rendering code and you've got a kick-ass portfolio. Awesome! But can you talk intelligently with the game designers, artists, narrative designers or writers on your team? Having some of the basics in these areas will allow you to communicate much more fluently with your colleagues and make you a better team member.
4 - Take part in the community
The game developer community has some of the most awesome people you'll ever meet. In my experience, game developers are smart, passionate and welcoming people. (Note that I am not talking about the gamer community). Get involved with you developer community. Attend events, organise meetups, and meet people. You'll enjoy it, learn tons, and build a network that will make finding a job so much easier.
5 - Make more games
The bottom line is that you only get better by doing things. Make a game, learn from the experience, repeat. There's no shortcut. There are always game jams and competitions you can enter, and these will always be fun.
Hope that helps and answers some of your questions on how to improve your odds of getting a development job!
Good luck, do let us know how you're doing we love to hear your stories and successes.