Hindsight being 20/20, I'd have made contact with the interested party to tell them that your resume was undergoing a major update, and would it be OK to send it in a few weeks. This would give the person the opportunity to write you back if the deadline is urgent. Often entry level positions and internships have very set deadlines and missing them can be a bigger problem than a less than perfect resume.
But submit it anyway. I often submit resumes with three parts:
- an email with a personal note
- a cover letter to the unknown main reader
- the resume
In this case, my email with the personal note would include a thank you, your hope that you have not missed a deadline, and a recognition that you are several weeks late. Don't try to hide it, but don't fall all over yourself apologizing. Skip detail - but you can say that you were working to improve the resume and getting feedback took longer than expected. I would avoid any excuses (got busy with classwork) as it implies an inability to juggle multiple demands, where honesty will serve you just fine.
But keep it terse - if it's more than 3 sentences, it's probably too long.
Then in your cover letter, say nothing of it. Stick to the good stuff - why you are awesome and why you want to work at this company.
The point is that you made a personal contact who deserves the courtesy of knowing why you kept him waiting. But your contact will pass along this information to someone more involved in the hiring process who neither knows nor cares that your resume was delayed. So make it easy for your contact to see the apology, but make it easy for your full application to go through the bigger system apology-free.