One thing you can look for is compare your own performance to someone who's doing better (you can sort players based on their job and DPS) in same fight and then look at things like your activity % and actions per minute (these show if another player is for some reason able to have more DPS uptime in the same fight or is using more abilities within certain time frame than you are). You can also see what their most used actions list looks like compared to you (if they for example use certain skill 10 times during the fight when you only use it 5 times etc.). You can even check the full time line of what someone who's performing really well does during the fight.

For me personally, my most important method for self-improvement has been taking videos of my own playing and then watching them afterwards. During the fights it's hard to notice things like (in my case as SCH/SMN) refreshing DoTs too early, doing unnecessary movement or not taking advantage of opportunities for instant casts while moving etc. If you have a chance to do this, I certainly recommend it in addition to using FFlogs.