As a fan of both (first Gaga, and more recently Michael) I think a lot of the comments here are unfair towards her.
She's always been a huge supporter of Michael in the press, and not just the usual 'he was an inspiration!' shtick. I believe it was her 2013 Howard Stern interview where she first spoke of buying huge droves of his clothes, keeping them expertly temperature controlled and one day using her own money to create a museum that fans could visit to see these costumes.
She's not someone buying these costumes only for the 'iconic' ones. She actually most of all wanted his tour costumes - ones that to the average Joe may not have seemed special, but as an artist herself she remarked that her tour costumes had the most sentimental value and she wanted to salvage these articles of clothing from being scattered across the world in the closets of rich businessmen who had no regard for Michael. When her grandfather was dying I believe she a jumper of his that she purchased with 'MJ' embellished to make her feel better, but she said she'd never dare wear anything the public would ever instantly recognise, out of respect for his legacy.
She has also said that she completely backs his innocence and believes he died, ultimately, of a broken heart. I hope people realise that she does have respect for him and isn't just doing this to be ~shocking!~ or controversial. It's all out of love.