Sometimes verb; more often adjective. Acronym for “do-it-yourself.”
Google Ngram shows roughly a doubling of American use since 1990 (with a bump presumably associated with the launch of the DIY Channel on cable television in 1999):
DIY also increased in Britain in this period, from about double U.S. use to triple–demonstrating, I suppose, the transatlantic power of cliched catchphrases.
Owen Wilson, not yet a bankable movie star, and Jackie Chan, not ready for retirement, return for a second round of D.I.Y. stunts and business-casual wisecracks in this sequel to “Shanghai Noon.” (The New Yorker, March 3, 2003)/Indie-minded artists, storytellers and comics creators have banded together to form a pop-up space called “Tr!ckster” that will celebrate the spirit of DIY and creator-owned work. (Los Angeles Times, June 21, 2011)
notoneoffbritishisms.com 
“DIY” has been a catchphrase in American punk/alt culture since like…the 80s? The second example is clearly in this lineage, it owes more to the slow-then-fast mainstreaming of indie culture than to Anglicization (though the two are kind of conflated I guess). Thinking of “Twee” as a britishism is problematic for similar reasons.
DIY was one of the core themes underlying British punk right from the start in 1975/6. So it would not be surprising if this transmitted to the US.
The phrase was the title of a great song by Peter Gabriel (not a punk) in 1978.
^ What anon29 said is exactly what I was coming here to say.
I find that Brits use “DIY” as a standalone noun much more than Americans, as in “Doing a bit of DIY on the weekend.” An American is still much more prone to speak of “a DIY project” or the redundant “I’m doing it DIY”. Also, the rise in both countries is directly connected to the rise of the big box retailers such as Lowe’s and Home Depot, along with the TV shows/whole cable channels and books and magazines. In other words, the terminology is following a real trend.
Used here also as ‘Destroy It Yourself’ which many people do and pay for the priviledge
In the 1980’s, i had to ask my Lancashire matey what “DIY” meant. But now, perhaps of all the Britishisms listed here, this expression is ubiquitous in the USA.
Really an initialism rather than an acronym – ie it’s pronounced “dee eye why” not “dye”.