Remember, you're hoping the person who receives your letter has no memory of your closing at all. The last thing you want is a hiring manager going into an HR meeting with your cover letter in hand, asking the team if they want to meet with "Mr. Kindest Personal Regards."
Personal pronouns (like I, we, and you) are important in letters and memos. In such documents, it is perfectly appropriate to refer to yourself as I and to the reader as you. Be careful, however, when you use the pronoun we in a business letter that is written on company stationery, since it commits your company to what you have written. When stating your opinion, use I; when presenting company policy, use we.