I learnt the craft on Critique Circle as well. I only joined about half a year ago, so I wouldn’t count myself among the pros.
Usually I read the whole text first and then don’t look at it for a day or two. Although I constantly think about it during my daily routine. And when I get back I read it once more and then give a critique. That way I am able to tell if it made a deep impression on me. I think that is an important aspect.
Since English is not my mother’s tongue I’m not the biggest help regarding grammar issues. But I’m good with typos.
I mostly concentrate on highlighting what doesn’t work and explaining why. Because I believe that is the main purpose of a critique: to help, not to praise. I used to forget to mention the good parts. Nowadays my feedback is more balanced. At least I hope so.