Hard to say what the problem could be.
I'm not sure how Steam's multiplayer works, but I have had issue with it before due to corrupt files, or games patching incorrectly. You could try verifying the integrity of the game files. Right click on the game in your library, go to local files, and click verify game cache files. If there are any discrepancies it will identify them and sort it out.
Another possibility is that you need to port forward via your router. I don't see why you would, as the game runs through Steams multiplayer service, but it might. Hamachi would help with that, but if you don't want to use it, you could try digging around for the ports used and manually port forward yourself through your router settings.
Honestly though, in my experience, Hamachi is very easy to set up and remove if need be. I'd give it a bash just to test the waters. Worst case scenario, you just remove it. At least you will know if it's a router / port issue or not.
Might be something stupid like hardware compatibility, or that your OS doesn't play nice with the game. I'm not tech wizard, I honestly don't know. These are just things that have plague me before, so...
I hope the problem is resolved for you soon, good luck, and once you get it working, have fun.
