I have just started taking a New Testament Greek class. And while learning the Alphabet I see there is no J. So how do they get the letter J in the name of John in the Greek New Testament as in Mark 1:4?

The name “John” is a translation, not the original. The Greek for John is Ioannes, which doesn’t have a J but an I. For the (rare) sound of J, usually appearing in words imported in Greek from other languages, the combination of T and Z is used.

[?] I don't understand what you meant by "the combination of T and Z is used" Does it make sound different? Would you elaborate some on this?

To pronounce the English "John" in Greek we would write Τζών, and there is no other way of transliterating it.