Neutral is a rly hard concept to grasp and I've also only recently started thinking about it during matches. I was basically in the same boat as you, though.
Some things that I've noticed have helped me:
Working on spacing and baiting things out with dash-dancing and then whiff punishing.
Focusing on controlling space and attacking the characters "hitmap" (the area around a character controlled by their attacks i.e
http://i.imgur.com/vjs046X.jpg)
Focusing on what types of attacks to use and when to use them (Check out SSBM Tutorials new video on the types of attacks and think especially about when/how/why to use them).
Be super careful about what options you choose based on what they have a habit of doing.
Watch top level players and take note of their movement/attacks and analyze the reasoning behind them.