If you want an apartment, CraigsList is the go-to place.
All I know about Judiciary Square is that it's where courthouses and federal buildings are (and the DMV) and one of the most mispronounced Metro stops within the five-line system. I've only utilized that stop once so I never got a true feel for the area besides that.
I am Black, but am not a Capitol Hill intern or employee there, so I couldn't help you with that one. I'm sure you will find a group or niche of professional Black people interested in the Hill, just keep looking.
Culture in DC? It's varied. There are different types of people here: Doctors, lawyers, interns, college students, people from different countries, hipsters, laborers, etc. And on the downside there are thugs, hoodlums and rowdy kids trying to ruin the livelihood of DC, but if you live your life and have street smarts, you'll be fine.
As usual, more will jump in with a lot more to say and good advice.
PS: If you really want a feel for DC culture and life, check out
DC Blogs. You can get a great feel of how DC's like reading it from the people who live there. A lot of different blogs are about DC fashion, nightlife, the different neighborhoods, gentrification, etc.