What should a Cover Letter look like for a non-target candidate and general PE advice

Hey guys, so I don't want to make this an introduction thread or anything, but I'm hoping to go into PE one day, but I feel like I am an atypical candidate because I am [vastly] under-qualified unless I really catch some fortunate breaks.

Quick bio: I'm currently a senior Finance major at a non-target (but a top-25 overall undergrad business school; top-10 among public schools). I have a very sub-par GPA (a little below 3.0). I don't have any jobs lined up for after graduation, but I am graduating a semester early in December (3.5 years). I've held an internship for four summers, but only one of those was even somewhat relevant to what I want to do (and that was my first internship, which was at a MM in the wealth management side of the company -- though I'm still looking at Asset Management careers as well).

As I was looking at various job postings sites, I came across a very interesting opportunity. It is with a small PE firm (only a couple years old), and they are looking for interns for a couple of months. While getting an internship out of college isn't exactly ideal, I think that this is the best-presented opportunity for me if I hope to one day transition into PE. I have no IB experience, but I am about to finish an M&A class here at school and have become fairly solid at modeling, interpreting the three financial statements, and understanding multiples, valuations, etc.

I understand that I'm not an ideal candidate, but what made this opportunity particularly appealing was that I didn't have to have any prior experience, and they seem to stress the importance of wanting to train their interns, which would provide a fantastic opportunity to learn more about PE and hopefully help me land a FT gig after or perhaps transition into another industry where I could gain further experience necessary to revert back to PE later.

With that being said, what should I include in a cover letter? Should I touch upon my low GPA and [somehow find a way to] explain why it shouldn't deter them from considering me? If anyone could provide any advice/suggestions for how I ought to approach the cover letter I would greatly appreciate any words of wisdom. Not to sound overly corny, but after reading the qualifications of the desired candidate it appears as if they stress the importance of the individual and his learning more-so than their background/qualifications.

I realize that it's almost seemingly impossible to break into PE without relevant IB or consulting experience, and I think it's too late to find something in that regard, but I'm an optimist and keeping my chin up. Thanks to all in advance.