Black racists will often accuse other black people of "acting white". There's even a rather old moniker "oreo" to describe this ("black on the outside, white on the inside"). Black people who "act white" are deemed as some kind of race traitors, because they don't "act black" enough, and thus are traitors to their "own people" and their "own culture".
But what exactly is "acting white"? It's actually hard to get concrete answers.
Deducing from everything I have seen, when they accuse black people of "acting white", what they mean is somebody who (typically) gets a higher-degree education, ie. goes to school to study hard and get a degree (and typically does not participate in any political activism), gets a good-paying job corresponding to his knowledge and education, performs that job diligently, using all of his knowledge, education, qualifications and experience, is perhaps promoted because of his job proficiency, gets a quite good income, buys a car and a house in a decent middle-to-upper class neighborhood, gets married, does not make a big fuss about politics and such, and overall lives like an honest, hard-working, law-abiding, successful citizen.
You know, like a normal person in a normal society, pretty much anywhere.
There are no traits, behaviors, opinions and characteristics that are common to all "white people" (whatever that term would even mean, given that it's not completely unambiguous; for example, are Spanish people "white"? Are Jews "white"? Are Brazilians "white"? And so on). Even if you took the "whitest" people you could think of, they are still extremely diverse in their traits, behaviors and opinions. There is nothing that's common to all of them, when it comes to behavior or culture.
When black racists talk about "acting white", what they seem to be talking about is the ideal modern western sociopolitical culture (which is, in fact, not restricted to the west, as it's very common in other parts of the world as well, such as Japan, India, and so on) of honest hard-working law-abiding citizens who just live their everyday lives and get along, without making a big fuzz about things, and overall have an live-and-let-live attitude, and a "we should all work together to build a better society" attitude.
When it comes down to it, when we really examine it in depth, when these black racists talk about "acting black", what they are talking about is, essentially, racial hatred and having a "gangsta" thug culture and attitude that hates white people and is very tribal and isolated, and hates the idea of a peaceful working society where everybody lives in peace and harmony, working together and working hard to earn a good living, and owning a good home in a good neighborhood, and living a peaceful and prosperous life.