Computer course with English, in Greece..?

You need to catch up with a frontistiro. All Greek kids now want the EDCL - exam. Better work with a frontistiro first, then move over to private lessons when you've learned everything about the exams etc.

That's a tricky question: Teaching is not well-regulated by Greek law, and whatever the law provides not always consistantly enforced. At any rate, there are two major areas you need to work on, financial registration and academic qualifications.

In theory, you would have to register with the local Revenue Service (ΔΟΥ), who should give you details about the book-keeping you need to do. You will also have to register with Social Security (TEBE), which may cost as much as 300 Euros per month for the first two years and more later. Unlike income tax, this amount is fixed and does not depend on how much you earn. This is why, in practice, many teachers who provide 1-on-1 tuition avoid registration.

I am not sure what academic qualifications are required for teaching computer courses, but a teacher of English should have a teaching permit (Άδεια διδασκαλίας) from the Ministry of Education. Το get one, you need to prove language competence (Επάρκεια), you have to pass a medical examination, you must have a clean criminal record, and -probably- legal resident status. After application, you should allow as much as 10 weeks for the permit to come through.

All this paperwork is probably necessary if you want to work as a free-lancer offering your services to private schools (State schools do not hire free-lancers). However, if by "privately" you mean that you plan to do 1-on-1 lessons, things are rather simple, because people rarely ask for any of the credentials above. This means that you can get away with teaching "informally" until you can sort out all the paperwork.

Good luck!