How do I know if I qualify to be a Law Enforcement Officer?

SPC (Join to see) Good question. Usually the departments you are interested in have websites that you browse the basic requirements. As SSG Bryan McNally related, they are looking for clean cut. I would add to his list and put financially responsible-lives within means, and has good credit. Think Military Security Clearance, and you are on right track.
If there is crap in your background, be prepared to answer for it intelligently and know that some things will be automatic D.Q. Most agencies will do a back ground on you.
In an interview, be prepared to talk about ways that your Military Service has prepped you for the career you seek.
And, know that PDs, SOs, Staties and Feds are full of veterans. It is likely that your interviewer will be a vet, or someone on the panel will be.
And, know that after a few years of Military Service, you HAVE qualities that departments want, you just have to rise to the top of the crop to get selected to advance the process.
*if you are not already done with college, get into it. That is definitely the way things are going.

You have to check the requirements, state by state, agency by agency, as they are all different.

Do you have any kind of police record in the military or on the civilian side? If not, then as long as you pass their requirements, then yes you can.