Hell, even I can come up with easy ways to include black people. I'm not advocating this, but thinking that Mad Men doesn't produce stories with blacks because they can't -- because the show is set in the 1960's -- is simply incorrect. It is because either AMC doesn't want it or Matthew Weiner doesn't or both.
As others have noted the show is ultimately about Don. Don has a black secretary but it's also clear that Don is far from the most enlightened person. In the first episode he was sleeping with a Jewish woman while bragging to Roger he hadn't hired any "on [his] watch."
Remember, the only reason he hired her was because: (a) he had an idea for exploiting the civil rights movement, resulting in a flood of unwanted job applications from black people; (b) Cooper and Sterling (whom you might remember parading around in blackface a party) figured making him hire her was comeuppance for Don's idea and/or for sleeping with and/or marrying the office staff.
As you point out, the show is Don's story. Don is not someone inclined to associate socially with African Americans. It would be a betrayal of the character to do what you want.
Don, who is cheating again, has an affair with a black woman--it turns into more than he expected. Roger has an affair with black woman... Pete has affair with black woman... More provacatively, Peggy has affair with black man... The agency gets into financial trouble, sees advertising to blacks as a way to quick bucks with no competition -- hires black person to culturalize creative staff. They could do a story featuring Don's secretary.
See above, re: Don.
As for other characters, Pryce had an affair with a black woman while his wife was in England. Kinsey also dated a black woman for a while. So what you want has effectively already occurred.
Furthermore, as touched upon above, Roger is an overt racist. Besides his antics in blackface he separates jews from "normal" people and made his feelings of contempt for the Japanese explicit to potential clients. He's more polished than, say, Archie Bunker, but about as unlikely to have an affair with a "Negro."
And, to be frank, Peggy isn't much better. While she felt guilty about it, she hesitated leaving Don's secretary alone with a large sum of money.
Ironically, of all the main characters, Pete is probably the least racist.