As a writer, I tend to think about media bias from a different perspective than the average media consumer, and also different from academic researchers who try to identify media bias via data analysis, as described in our recent podcast "How Biased Is Your Media?"
I tend to think about subtle but telling things like word choice and sentence structure -- what is the journalist emphasizing, or downplaying, and why? -- but also an article's placement, inclusion or exclusion of outside quotes, and choice of headline (which, for the record, is usually written by an editor and not the reporter him/herself).
Above all, I tend to compare articles from different newspapers that are based on the same event. This is to me one of the simplest but most powerful ways to take the pulse of a newspaper's culture. If, for instance, two newspapers publish articles based on a simple event -- a state comptroller's report about Wall Street bonuses, for instance -- one can read a little bit of institutional attitude into the two papers' resultant articles.