Cloning animals isn't actually that hard. We did it in the 90's with Dolly IIRC. There just isn't a whole lot of practical use for it apart from human cloning, which is illegal.

It's pretty straightforward: you take an egg cell, suck out its genome and replace it with a new one. Then you implant that egg in a host and it is gestated and born in the usual way. The tricky part back then was extracting the from adult cell samples, but modern genomics is way past that. It's pretty much the same process as creating a new bacterium, which is done quite commonly as I understand.