It depends on where you're applying and what concentration you're applying for. I have a non-science background except for some first year classes. I got in everywhere I applied (including Columbia, Emory, Pitt) except for JHU and Harvard. MPH programs are not all that competitive.
Below is what adcoms are going to consider competitive:
1. Strong statement of purpose; I linked my experiences to what I wanted to do in public health. I talked about why I wanted to make the change and how my skills are transferable.
2. GPA over ~3.0; ~3.75 for funding from the top schools. My GPA was 3.75 (SOPHAS had 3.6), I got some funding but yours is higher so you should have no problem.
3. GRE scores >75% to get in everywhere, >90% to get funding everywhere (much lower for anything that isn't a top 7 school). I had 95%V, 71%Q, and 98%AWA for GRE scores. I think I could have potentially gotten more funding offers if I had re-taken and increased my quant score.
4. Experience? For some concentrations it is highly desired or mandatory, depending on the school and concentration. Some don't care at all. Check on the admissions websites.
5. Pre-reqs? DON'T waste your money on these. Honestly for HPM you probably don't have to take any, but I have no idea since I applied to epi. They might want you to take intro to bio at some schools and have stats, but decide where you'd like to apply before spending money on this because most really don't have any requirements.
6. Rec letters - just make sure they're from people who like you.
This is the stuff I wish I had known before I applied coming from a business background. Don't sell yourself short. Apply to as high of a "reach" as you want to go because, I'll re-iterate, NOT THAT COMPETITIVE!!!. What IS competitive is funding. So, while you will get in pretty easily, you want to make sure you get your scores as high as possible for a fighting chance from the top schools. You have a high GPA. It sounds like you're motivated and probably pretty smart, so you'll do well on the GRE if you prepare for it. Write a good essay and you'll be fine.
Don't stress the app process as much as I did. What you really need to plan ahead for is the move if it's going to be a big one...
Good luck!
Edited this because I just saw you put HPM in your original post...