Hey there! I'm actually a CS student in my first year. I haven't actually been employed over here but I do help out at a couple places and am working on some projects with friends/startups.
It's a lot easier for me since I moved from Luxembourg and German is one of our official languages but I'm pretty sure you should be fine if you have decent knowledge of the language or just get a job at a more international company. There's LOTS of startups here in Berlin for example, with most people just speaking English because why not.
I assume some companies might want to know who they're working with so proof of studies is always a plus. If you can, take tests that give credibility to your skills. I know for a fact there are a couple of services here in Germany that allow you to take tests and hand you certificates.
Living here in Berlin as an English speaking person is fairly easy. I know a lot of people like that and most officials do, too. (I know a couple french people that only speak french and broken English and they are fine.. but their work mostly consists of delivering food or something like that, but there are also programmers among them)
The most important part about moving here is actually knowing if you
This is just for Berlin and from the perspective of an EU student.
If you don't have a plan and just move here thinking you'll have an easy time finding a job with a couple bucks in your wallet you might fall hard. Unless you don't mind picking up jobs you're obviously overqualified for or move here without having found a position first (there's a couple pages where employers actively look for people, but I can't vouch for any of them) you should probably stay away! Definitely not a cake-walk.
If there's anything more specific you want to know feel free to ask!
P.S. There's pretty much always people looking for competent programmers. The startup scene is amazing here in Berlin. Mostly just a matter of knowing where to look, or just being at the right place at the right time :)