You should use the possessive adjective, "my."
The promotion is the object of the verb "led." If "me" were the object of the verb (e.g., "The trail of evidence led to me.") then you would use me, which is an object pronoun. Since the object of the verb is "being promoted," use the possessive adjective to indicate that it's your promotion.
Usually we don't make this distinction in spoken English, and it might sound stuffy to use the possessive pronoun among friends. But in written English, and especially on resumes or in business documents, the possessive adjective is definitely preferable.
Here's a more thorough explanation of the general rule.
I agree with Ben, though, that it's simpler (and better) to say "my promotion" rather than "my being promoted," and if you can phrase your promotion as something that you brought about actively, rather than passively, then all the better.