"Blackfishing is when White public figures, influencers and the like do everything in their power to appear Black."

"Whether that means to tan their skin excessively in an attempt to achieve ambiguity, and wear hairstyles and clothing trends that have been pioneered by Black women.... Instead of appreciating Black culture from the sidelines, there's this need to own it, to participate in it without wanting the full experience of Blackness and the systemic discrimination that comes with it. " 

Said journalist Wanna Thompson, quoted in "What 'Blackfishing' means and why people do it" (CNN).

Leslie Bow, a professor of Asian American studies at the University of Wisconsin, describes Blackfishing as "a racial masquerade that operates as a form of racial fetishism.... They might think that it operates as an homage because it appears to honor Black style. In this case, they graft off of what academics have called the esthétique du cool that attends Black culture.... Blackfishing is one form of racist love, how we appropriate otherness."...

"People have no respect for Black culture -- they think it's up for grabs and will take and participate in it and exploit it because it has been commodified," Thompson said. "We see this on every social app, but more increasingly on TikTok." But when people change their appearance with makeup, cosmetic surgery or digital editing to appear Black, it goes a step further than cultural appropriation. It's Blackfishing.

Here's an example of something that is said to represent "blackfishing" — Iggy Azalea, "I Am The Stripclub." I consider that so on-its-face racist and sexist that it's hard to formulate why it is so awful. I'd say the problem with calling it "cultural appropration" is that no one ought to want to say that looks like my culture. To the extent that it deploys offensive stereotypes of black people, the traditional term is minstrel show. 

But I think the term "blackfishing" more specifically relates to the makeup Azalea is using — makeup and maybe also hair and clothing. It's fashion and beauty. That appropriation seems to be entirely complimentary to black people, and the criticism has to do with reserving this look for black people.

The "-fishing" part of the word "blackfishing" comes from the term "catfishing," which has to do with presenting a false persona on line. Wikipedia explains how fish got involved: 

The modern term originated from the 2010 American documentary Catfish. The documentary follows Nev Schulman, the executive producer, as a victim of catfishing. He had cultivated a relationship with what he thought was a 19-year-old girl from the Midwestern United States. The woman with whom he had been communicating was actually a 40-year-old housewife. In the documentary, it is mentioned that the woman's husband came up with the term as he found comparison to a myth how cod and catfish act when shipped in different tanks. The myth describes how live cod fish were shipped with catfish in the same tanks to keep the cod active, ensuring the quality of the cod whereas being shipped alone the cod would become pale and lethargic.