I can't say from personal experience since I haven't had to write a profile on anyone in the time when social media backgrounding became a common thing. In terms of what things to search for, I imagine it's all subjective, just as it's subjective on what you should judge someone for (social media, criminal background, etc), but I'm sure looking for common bigoted slurs would be standard practice.
One thing worth noting: As I understand it, Mr. King's charity campaign was heavily based off of social media (after the initial appearance on ESPN GameDay) – meaning that he spread it via his own Twitter account. Which makes looking at his Twitter account and past tweets more routine, since social media is essentially a large part of his current fame/notability. For other kinds of profile subjects, such as "Teacher of the Year" or "veteran recalls memories of war on war's anniversary", I'd be surprised if reporters did a social media check. Because unless that person themself says their social media profile is a big deal, then the reporter probably won't even be aware of it.
One thing worth noting: As I understand it, Mr. King's charity campaign was heavily based off of social media (after the initial appearance on ESPN GameDay) – meaning that he spread it via his own Twitter account. Which makes looking at his Twitter account and past tweets more routine, since social media is essentially a large part of his current fame/notability. For other kinds of profile subjects, such as "Teacher of the Year" or "veteran recalls memories of war on war's anniversary", I'd be surprised if reporters did a social media check. Because unless that person themself says their social media profile is a big deal, then the reporter probably won't even be aware of it.