I recently brought a large crate into my room for my Dachshund puppy so she would have more room when I was away. My cat keeps going in it, scratching around and hissing when my puppy gets near it. I have a feeling that he is trying to "claim" it. I have been removing my cat from the crate and saying "no", and putting my puppy in it and praising her. I have been leaving it open so she is free to explore it today. Am I doing the right thing?

Find kitty an open cardboard box for his crate. Lay it so the open side is on the side, not on top. (Cats love caves.) Put a soft small blanket inside (warm it first in the dryer and let the cat snuggle into it).

That's a good idea.
I swear to god they are like brother and sister.
They both compete for everything, but love each other to death.
It's so cute cause they wrestle all the time and cuddle.

Who said dogs and cats don't get along?
Haha

If kitty sits on top of the box, it will eventually collapse or at least sag. Pet stores have similar cat dens made of more durable material, or just find another cardboard box. Xerox/copier paper boxes are strong-sided, and reinforce it with duct tape etc.; check with your public library or Kinko's for one.