Yo escribo una carta is right, but you are only saying 'I write a letter', right? About the second one, Yo te escribo una carta a ti is wrong, the 'a ti' is replaced by the 'te'. Yo te escribo una carta would be better. Still, we try to avoid the redundancies in Spanish, and the person (I), is clear because of the conjugation of the verb. So 'Te escribo una carta' would be the best for `I write a letter for you'. Take care, Alejo.

Te escribo una carta