The normal way for groups to join the system was to organise your group locally and sell their votes to candidates, later you ally with a party and some of your goons get no-show jobs, later you become part of the party and your people move into low level positions etc
This worked for the germans, the italians, the irish and other european groups, slowly but surely.
Of course back then local and state government counted for a lot more than federal
This hasn't worked so well for blacks, oh the black upper middle and upper class is doing well in the burbs, but...
It also appears that some black leaders are becoming disillusioned by their experience as a wholy owned subsidiary of the democratic party.
It basic problem is that they have become a safe voting block breaking 90+% for the democratic candidate no matter what - and safe votes are free.
There's no particular reason such a party couldn't work, it just requires a very different approach to the system.
It requires a group that is large enough and geographically concentrated enough to have a significant base of "safe" localities incl some state & house seats.
It also requires a high degree of internal identification and willingness to follow a group decision once made so that the group/party reliably can sell the vote with a high predictable turnout.
In areas where you're competitive you run serious candidates, in those where you're not but can still decide the outcome you run "resigners" while you negotiate WHICH - if any - candidate to back and for what price and in hopeless districts you run for PR and attention to your particular problems & interests (that's where you put the Jesse Jackson Jrs, Al Sharptons and other hotheads).
It does require hard nosed leaders willing and able to prioritise their group/constituents above personal interests and thus keep the backing of the group.
It also means that even if you regularly ally with another party you MUST be willing to occationally sink their candidate - even for president/senate/governor.
You HAVE to be willing and able to deal with all sides, otherwise you end up with a buyers market and get the same as OTL - bupkus.
Obviously such a party will never win the precidency, but may still influence the election and so gain for their support - the bargaining will just be up front and more publicly visible rather than the usual smoky backroom thing - and again you have to be willing to say no/switch candidate occationally to keep your bargaining position.
It may well come off as rather kras and is unlikely to win you many friends, but if succesful the rewards can be significant
Blacks are a large enough group and sufficiently concentrated that a safe base can be established locally and in some states.
Whether internal cohesion and leadership is up to the job is a different matter, it would probably have helped if the black middle class had stayed near the rest of the group. You probably need the black churches deeply involved to keep enthusiasm, commitment and turnout up.
It is also likely that such a party would be much more socially conservative than OTL democrats - no gay marriage support or transgender rights there.
Of course the democrats would fight this tooth and nail, it's not just the black vote, if they manage it latinos will be next, and the democratic party would be well on its way to being a party for students, government employees and liberal city areas similar to the Lib Dems in the UK (slightly exagerated, but...)
As for when and how, it has to be well after the civil rights era, otherwise you probably get a bi-partisan reaction that blacks really aren't ready for democracy yet and it may be better to keep things like literacy tests etc
You likely also need decades of experience to build up af large group of people with real experience of how the system actually works behind the scenes and where the real barriers are etc. Also something like the disaster that drugs and thus the war on drugs has been to shake up trust in the system.
Best shot for a POD:
The 2008 dem primary race gets MUCH nastier with Clinton squeaking into first place using every dirty trick in the book to grab the last few delagates.
Black turnout is significantly lower than normal and Clinton narrowly wins in november. Notably the democratic vote is significantly depressed in states with many black voters leading to a few unexpected losses and some squeakers.
The Clinton administration now don't feel they owe black voters anything and relations with the Black Causus deteriorates
By 2010 a Black Party runs in many local and some state elections in districts with many black voters - effectively against white democrats and black incumbents seen as corrupt / incompetent / too "machine" and scores a number of wins.
By 2012 they run in most races in districts with large numbers of black voters and grabs some house seats, numerous state house/senate seats.
By 2016 they run a presidential candate who withdraws after the final debate and declares for the dempocratic candidate.
In other news a number of democratic candidates in house and state races withdraws in favor of the Black Party candidate...