Software Engineer at Google
Being a Computer Science student is a good start. Being a Front End Web Developer is a completely reasonable specialization with that degree.
The main thing you need to do is, of course, be able to write Web pages. I don't know how much you know, so I'll go from the basics even though some of it might be really obvious to you.
You might be able to take classes that are based on Web development, or may need to do it yourself. The Mozilla Developer Network, developer.mozilla.org, has a lot of excellent tutorials and references for modern Web technologies. Even if you are able to take classes, I recommend making some Web pages yourself. The experience is very valuable, and will help you talk to potential employers as someone who is well-versed in the Web instead of just having taken classes on it.
I hope this helps!