How does one become a programmer for the gaming industry?

Develop High Level Math Skills. ...
Get a Bachelor's Degree. ...
Build a Portfolio. ...
Find an Entry Level Position as a Game Programmer. ...
Advance to Become a Software Developer.

First, you need to have interest in game development. You could possibly start from your own indie games very early on, whether plunking away on any IIc or on a TI-85 calculator.

There is no need to have any artistic skills, so you can learn some arts from other games just so you can achieve progress on your work. One of the great joys about being a game developer is the cross-disciplinary experience of it: and you'd get to work with some of the most talented artists in the world!