Yes, the "Black" in his name was a reference to his race. Black Lightning was (arguably) DC's first black superhero. When recounting the story of his creation, Tony Isabella (the writer who created the character with the artist Trevor von Eeden) notes that DC was considering creating a super-hero who (if I recall correctly) was a white guy who turned into a black guy in his super-hero identity. Tony didn't think this was a good idea, and came up with an alternative.
Black Lightning (in the comics, at least) is Jefferson Pierce, a former Olympic athlete who came back to his old neighborhood (a part of Metropolis known as Suicide Slum) as a high school teacher. His father had been killed when he was younger, and he had a desire to clean up crime in the neighborhood, to try to protect today's kids from the difficulties he faced.
Note that the character's original costume included an Afro wig (Jefferson's actual hair was much as seen in the image in the OP, from slightly later in his career). A large number of his students were black or Hispanic, and he wanted to be an example to them, as noted above.
As far as how he got his powers: Initially, he had no super-powers (although, again, a former Olympic athlete). He did have a friend and mentor from his youth, Peter Gambi. Peter (unbeknownst to Jefferson) had both a criminal background (he was actually responsible for the elder Pierce's death, as I recall; he regretted what that did to the man's family, and tried to help them, becoming their friend in the process), and a brother who (at that time) was still involved with criminals; Paul Gambi supplied the costumes (at least) for most of Barry (the Flash) Allen's rogues gallery. With these connections, he not only created the Black Lightning costume, but supplied the belt which gave Jefferson the ability to generate electric shocks (hence the name), and provided a force field.
The belt somehow transferred the powers to Jefferson himself after a time; it's not clear whether this was an intended outcome, or the result of an accident. Later stories have specified that Black Lightning has the "meta-gene", which allows DCU humans to obtain internalized powers, but the early stories leave some suspicion that the belt was intended to give the powers to Jefferson over the course of time.
A further note on black characters with "Black" in their name.
From the 1970s well into the 1980s at least, it was very rare to introduce a "Black" character who wasn't black (it may have started as early as the introduction of the Black Panther (who wasn't African-American, hence my use of the less preferred term)). Prior to that time, there were a number of white characters with Black names: DC had Black Canary and Black Hand; Marvel had Black Marvel and Black Knight. After that time, new characters (Black Racer and Black Lightning) and revivals of old characters (Black Angel in Eclipse's AIRBOY) were almost never white folks (in the case of revivals, in spite of the race of the previous holder of the name).
Marvel's Black Cat is the only notable exception I can think of (and, in many ways, she was intended to be to Spider-Man what Catwoman was to Batman, and just needed a name that fit (and it's not like even Marvel was progressive enough at the time to put Peter Parker in an inter-racial relationship (Iron Fist and Misty Knight were a thing already, but Peter Parker people would pay attention to))).