I have three basic edits. I tend to get the spelling and grammar errors sorted out first (there are always some, especially if I'm writing quickly). I don't dislike doing this but it is my least favourite editing task so I am glad to get it done. I also check for context here. Spell and grammar checkers are notoriously bad for not being able to pick up on that. I see them as useful tools but a guide only. Nothing beats the human editor (and I can think of two people I know who would say "quite right too" - my editors at Chandler's Ford Today and Chapeltown Books!).
The second edit is for checking the story or article makes sense, that it is conveying what I want it to (and in the case of characters, they are coming across the way I intended. This doesn't always happen, sometimes the characters surprise me and I discover aspects to them I hadn't initially thought of but developed as I was writing the piece. I always check out where those developments go - they are almost always better than the initial idea but I have had to write myself into the character more to discover these developments!). But at the end of this edit, everything has to make sense. There must be no loose ends.
The third edit is "bringing it all together". This is where I check any additions to the plot/piece make sense, blend in seamlessly (so only I, and possibly my editors, will know those bits were added in and where!) and that I am happy to submit it. (This involves a final spelling/grammar check but the bulk of this was done on the first edit so this aspect here does not take long and acts as a failsafe for me to make sure I didn't miss anything).
Oh and I always, always, always edit on paper. You miss things on screen. I have no idea why it is but just know that's the case. Besides I think there is a pyschological element here in that when editing on paper, it feels like "proper editing".