Ok, so I have no drugs, no tickets, no criminal or bad credit history, etc.
What worries me is that I have a dual citizenship; I have American and
Russian (born in Russia) citizenship. I lived most of my life in the US and I am in my mid
twenties. Could this be a problem? My parents just want me to have it
just in case I want to travel to Russia for sightseeing and stuff.

I just received my packet with a bunch of applications and personal history statement
and it looks like my local department will be inviting me for a polygraph.
I know the economy is bad, but I heard that my department will be hiring 30 to 40
officers this fall. Apart from the economy, what chance do you think I have taking
into account my dual citizenship. BTW, if they ask me on a polygraph if I am
a spy or something like that, then I will definitely say NO with confidence which is
the truth.
I just really hope this wont be an issue.

Having dual citizenship shouldn't be a problem for you. Just so there is nothing suspicious relating to this in your background.

Jenn is right. Do you speak Russian? If so, that could help you get hired if there is a number of people that sepak Russian in the jurisdiction that you want to work.