* Είμαι ο/η τυχερός/ή. (Eimai o/i tycherós/í) - This is the most literal translation. Use "ο" (o) if you're male and "η" (i) if you're female. This is straightforward and generally understood.
* Είμαι πολύ τυχερός/ή. (Eimai polí tycherós/í) - This translates to "I am very lucky." Adding "πολύ" (polý) emphasizes the luckiness.
* Εγώ είμαι ο/η τυχερός/ή. (Egó eimai o/i tycherós/í) - This emphasizes "I" am the lucky one, contrasting with others perhaps.
* Μου ήρθε η τύχη. (Mu írhe i týchi) - This translates more literally to "Luck came to me," and implies that you're the recipient of good fortune.
The best choice depends on the context. For most situations, Είμαι ο/η τυχερός/ή or Είμαι πολύ τυχερός/ή will be perfectly suitable.