Officer Delinko reveals something about his past and something about his integrity when he asks Mr. Eberhardt to write him a letter of praise for his personnel file.
By asking for the letter, Officer Delinko reveals that he is willing to use the concept of "it's not what you know, but who you know that's important" to his advantage. Delinko wants to be detective in a bad way, and he is willing to look for shortcuts to get there. That doesn't make him a bad person, but it does show that he is willing to use other people's influence to further his career instead of personal hard work and diligence.
The specific part of his past that it reveals is that he is the officer that fell asleep at the construction site. He was there investigating something on his own time, which is good, but while asleep in the squad car, it was spray painted.