I'm in the same boat as you: I'm finishing university soon and I want to be able to work right away. I can't tell you how to make this work, but I can certainly tell you what I'm doing!
First, I made a bunch of games. Start small, make something with a tiny scope. Participate in game jams, and try to figure out who's "winning" and why. I was lucky enough to have my first game scored highly in the Ludum Dare and the result was some articles on blogs here and there. It was a tiny interactive fiction! Like I said, start small.
I started blogging and tweeting about other people's games. My feeling was: if I put the effort in now to spread the word about other people and their dreams, then when it is my turn to say "hey everyone go vote for me on steam" or whatever, I won't just sound like some rando. This isn't tricky: get a blog and post a few articles, throw up some links to your work. Get a twitter and don't just use it to say hi to people: post links to games you liked and mention their authors.
Number 1 on my list, though, is to find the local community. The city I live in (Vancouver) turned out to be an amazing place to make friends with other indies. Your city will too, no doubt! Go to networking events, like a local gamedev meetup. It can be really hard to talk to strangers (I certainly find it painful and frustrating) so I get around this by trying to always bring a game I've made to the event. Sit down with your game and your beer and people will come to you! Being at the centre of attention is often a lot easier than trying to navigate its fringes.
Don't stop there! Try to plan your moneys so that you can make it to some big industry events. I'm pretty poor, but I make sure that I can make it to PAX every year. Then I just wander the indie megabooth trying my best to say "hi" to folks. Remember to say "Hi, I'm Bob and I make games too!" rather than just "OMG U R HALF OF VLAMBEER". Folks will remember you if you introduce yourself and maybe ask them a question they were not expecting.
This year I also went to IndieCade and met a ton of awesome people! That was really hard because, unlike my local event where I am showing games, I wasn't showing any games. Because I spent a bunch of money on airfare, I felt a lot of pressure to "network" rather than just hang out, and as a fairly introverted dude that made the event pretty stressful. It can be tough, but it is certainly worth it to get out there and make some friends.
Yup! So, I haven't "made it" yet or even graduated, but I intend to be making games for a long while and this is how I've gone about it trying to make sure it happens.
hmm...
It might also be worthwhile to consider your alternatives: you don't have to work in the games industry to make games. I kind of want paid vacation and a reasonable work-week (like, you know, 40 hours) and I've been told it is tricky to get these things in the games industry. Your degree will make you an asset in most kinds of computer science, and web development is similar enough to gamedev that the work is often rewarding in itself. I've worked as a freelance web-developer throughout my education, and now that my time at school is almost done, I'm seriously considering staying. That way I can keep working on the kinds of games that you might find on SoftChambers, rather than the kinds of games that you might find on Kotaku.Ahem! Yes! Gamedev!
(also, post regularly on TigSource: this is something I fail at pretty badly)