As someone who is going to be in law enforcement, you should be required to undergo strict background checks to guarantee that they are hiring honest, safe, and reputable individuals in order to protect the people of their county. Professions that require you to have an open carry gun and involve you using it should not be taken lightly.
A police officer is essentially acting as an agent of the government: enforcing government laws, and protecting citizens. It is important that police officers have the highest moral character, and that they have a clean background as well. Corrupt law enforcement officers could violate the rights of the people they are sworn to protect. In short, yes, police officers should be go through stricter background checks.
Police officers have a special role in society. They have more power than the average citizen. They have authority over others. Because of this, police officers need to be of the highest moral order. Allowing people with tendencies toward violence or other issues makes everybody far less safe. Background checks need to be stronger.
How many individuals are willing to put themselves in harm's way on a regular basis, for a salary and a pension? These people have to be driven by something, and it isn't just a desire to do good, there are too many safer ways to do that. Under some conditions, they may seem less than sane to non-police. It's a hazard that is inherent in the job. Tightening up background checks will simply eliminate more candidates for a crappy job.