I'm only just learning to code myself, but CVs passed through my hand for several years at a moderately sized (50+) company. Most CVs went into a storage box or archive directory, never to see the light of day again.
Research the studios that you want to work for and find out how they break down their coding tasks.
First, learn C++. In the industry it will be absolutely necessary. If you only know how to work in C# and a job opportunity arises, you don't want to miss out. If you plan on working independently long term, learn whatever works for you.
If you don't have strong 3D and graphics coding skills and you apply for one of those positions, expect it to be discarded. If you do have strong skills in those fields, absolutely include a demo or a link to your website showing those skills/projects in action.
But you don't have to have leet 3D-fu skills to get picked up. Tools programmers work very closely with the designers and artists, and you get constant feedback on your work. If you like working with people and your portfolio shows that you developed game design tools this can be a rewarding experience. If you don't handle criticism well, it's not the role for you.
Sound, animation, scripting - depending on the project you may find positions dedicated specifically to those fields, especially if there's an engine in development or heavy modification of one.
Unlike 10 years ago where a bit of enthusiasm and just showing up with something cool could land you a job in-house, now roles are quite specialized. If you settle on one or two that you find exciting and hone those you can expand your skills as you go.
Additionally, expect a coding test. If you walk in off the street for your interview and show off something impressive, it doesn't mean it's yours. They will want to see what you can do. You can expect 3 or more hours of problem solving or code refactoring to satisfy your interviewers.
Good luck :)