Just as people are all unique, so are the ways copy editors approach a project. For example, some are more comfortable editing the "old school" way -- with paper and pen -- while others are comfortable editing right on the computer screen.
Personally for me, I used to prefer to go "old school," but have trained myself to edit right on the computer screen. At first it was challenging, but with time it's now a breeze.
Nowadays, when I first receive a document, I like to do a once-over and scroll through the whole document to get the big picture. I like to know what I'm going to be working with and mentally take note of paragraphs, headers, footers, headings, font, etc. I don't make many changes at that time unless it's something that jumps out at me.
Once I am done doing my once-over, I get to work starting from the beginning and working my way steadily to the end. All along I am reading for tone, readability, and consistency. I am looking for typos, misspellings, and incorrect information (for example, a type of technology or a company name that is misspelled).
As I am editing, I make my changes right on the computer screen. I use the MS Word feature, Track Changes, to show what I have changed.
Once I have completed my thorough edit, I preview the document using View Multiple Changes in Word. This helps me find any inconsistencies in the format. I also run a spell check.
When it comes time to deliver the final, edited product, I deliver both a version with the Track Changes turned on and one with them turned off. That way my client can see any changes if he/she want to or can just have the polished, finished product to work with.
I always welcome questions on why I changed something and I also always leave it up the author to reject any changes he/she does not agree with. Afterall, the client has to feel comfortable and has the final say.