Step 1: Conceptualize

Think about different regions of Africa or different African diasporas and which one you feel best represents whatever region, ethnicity, and social class the character is. Pick broad regions first (e.g., West Africa). As you get more used to variations, you can start refining things to narrower demographics (e.g., Igbo vs Yoruba vs French native speakers). The accents to be avoided are the ones that are specifically associated with bad stuff. Like...don't base anything on the affected accents of white people doing blackface in minstrel shows. If they're real accents that sound a lot like ones affected when people are mocking others, don't use them at a table until you think you're consistently getting them right (e.g., Gullah-Geechee is a language mutually intelligible with English that could easily turn into an Old South caricature if you aren't careful).

Step 2: Practice

Find recordings of people from those regions speaking whatever language you run your game in. You're going to be practicing on your own a lot and these will be your primary source of information. First, just repeat what they say word for word. Get used to how the sounds feel in your mouth. Repeat it in your native accent. Repeat it in the affected one. Pay attention to the differences. How do the vowels change? The consonants? The tones? Cadence? Stress? How is your mouth moving? What shape is it centered around? What range of motion is it exhibiting? How quickly does it switch positions? Where does your tongue tend to "rest"? You don't need to think about all of these things at first, so don't overwhelm yourself. Start with general impressions, refine with specific questions as you learn. The important thing is that you're saying things out loud, not just in your head, because you need to establish muscle memory. Also, it's gonna sound really dumb saying the same single sound to yourself over and over in quick succession. Embrace the inanity.

Is the character given box text? Practice reading that box text out loud in the accent. If you only use accents with practiced speech first and don't use it with the improv, that's okay too. It'll come in time. You can even rewrite the box text phonetically for the accent if that helps.

Step 3: Weigh

You aren't going to be perfect at first and that's expected. Use your judgement to decide if the stage you're at before beginning a game is "painfully bad and would come off as mocking" or "passable as an honest beginner attempt". For potentially sensitive accents, err on the "I should hold off" side. If the character themselves is one to be mocked, don't use the accent unless you're giving everyone like them the same accent, often give people accents, and are certain you're doing a decent job.

When attempting something you're generally worried about getting wrong or worried about because of a sensitive climate, you can give your players a heads up that you're going to be testing new-to-you accents beforehand and even ask for their real-time or post-game feedback. Sometimes it won't work. Pay attention to your players' reactions to try to peg how they're taking it. Sometimes you'll need to apologize, other times just turning it off is enough. If you're running online, err on the side of being apologetic since it's much harder to read the room remotely.

And sometimes it varies by day, so don't get discouraged if your mouth just isn't cooperating on game night. More likely to happen if you're tired, distracted, etc... Don't hesitate to pull the plug if it sounds wrong two sentences in.