Translation student here. Besides the obvious "TL note: this is X accent/dialect that is usually seen as Y-ish" route, it's probably exactly how you'd imagine you need to do it.
(a LOT of stereotypes in the text below)
Ways people speak languages can carry a lot of information about that person. Their birth area, their upbringing, education... So first you'd probably want to boil the dialect in question down to its various perceptions in the language's own world, at this point without keeping what you need to translate in mind, just looking at the dialect as is. That often leaves you with a lot of different, sometimes conflicting stereotypes, like how American Southern accents can be seen as old-timey, redneck-y, Civil War-y, even classy, among other things.
You then pick out which of these are the intended effects in the text (if there indeed is one), so a character speaking a Southern accent while wearing denim and suspenders and loading his revolver is clearly speaking that for a different reason than someone cleaning their musket, or someone wearing red checkered shirts and sharpening an axe.
Then you try to find alternatives in the language you're speaking that fits as many of these perceptions you've picked as possible. Compare using a Stalker bandit-like "cheeki breeki" Russian dialect and Imperial Russian to depict two different characters speaking a roughly similar "Southern" American English, one a rude hick and the other a 19th century officer.
If there is no direct equivalent (which there usually isn't, languages are not directly interchangeable), you rank the selected perceptions by their importance to the text and throw them away in ascending order, until you land on something that's closest.
At the end of the day though, it is entirely the translator's call, as there is almost never a situation (especially in fiction text) where something in one language can have a definitive, 100% aligned equivalent in another. It is always a compromise and a decision between many, many options a translator is faced with for every bit of text.
