Learn French: À quelle heure est-ce que tu te réveilles ?

tu is the subject ("you"), and te is the reflexive pronoun (which translates to "yourself" here). The dictionary form of the verb is se réveiller (to wake up). In French, you literally say "you wake yourself up". Since the subject is tu, the matching reflexive pronoun is te.

Where exactly does the reflexive pronoun (te) go?

It goes directly in front of the conjugated verb. The order is almost always Subject + Reflexive Pronoun + Verb. That is why the sentence uses tu te réveilles rather than putting the te at the end.

Do I have to include the À at the start of the sentence?

Yes! In English, we often just ask "What time do you wake up?", leaving out the word "at". In French, you cannot drop it. You must always say À quelle heure (At what time).

Why use est-ce que in the middle of this question?

est-ce que is a handy phrase that turns a statement into a question. When you have a question phrase like À quelle heure, you put that first, add est-ce que, and then keep the normal sentence order (tu te réveilles). This is a very common and natural way to ask questions in French.