。
She gave me an album as a birthday present.
Literal
She [topic-は] birthday present [as-に] album [object-を] gave-me.
The benefactive verb くれる is the giveaway that the recipient is the speaker (or the speaker's in-group). Japanese has three 'give' verbs that depend on direction: あげる (out from speaker), もらう (receive), and くれる (in toward speaker). No 私に is needed — the speaker as recipient is built into the verb. 誕生日プレゼントに uses に in a 'role / as / for' sense — 'as a birthday present.' This に commonly attaches to nouns naming a purpose or function.