Do not attempt to start working without permission from the German government. You’ll need a qualification such as a TEFL Certificate, Bachelor’s Degree, Associate’s Degree, or government-issued license. There are lots of online courses to get those basic qualifications.
If you’re studying at the University, there might be limitations on how much you can work. On the upside, getting permits when you’re already a student might be easier than starting out as a tourist. Some answers can be found here.
If you already have permission to live and work freely, make sure you’re not violating any laws by officially calling yourself a “teacher” when you are a “trainer”, “instructor”, or “coach”. It may seem unimportant to you, but it’s best not to confuse freelance English teaching with the “Englisch Lehramt”. Also, don’t work “under the table” or try to get out of paying taxes. Students can write off English lessons on their tax forms, and it will be cross-referenced to what you declare on your forms. Discrepancies will lead to an audit, and you’ll be in hot water with the tax authorities as well as your students.
Just to be Safe:
An important thing to get is First Aid Certification. It might be in your best interest to take a weekend course at the Red Cross and familiarize yourself with basic first aid, especially if you’re planning on being responsible for a group of children for any amount of time. Get familiar with the emergency numbers and other everyday bits of information that come in handy when you least expect it.