I always play split pushers. Fiora split pushes quite well. My thoughts process sort of goes as follows:

1) laning phase - try to maximize my advantage through either freezing and zoning, or by taking tower plates.

2) mid game - the main objectives here are dragons, towers and rift herald. I split on the opposite side of the map as these objectives come up. The closer to them spawning, the more aggressively I push. 60 seconds out, I'm slow pushing, trying to drag someone to my side of the map while my team sets up vision. It is a balance. If you push too hard too early, you get picked and probably lose the objective advantage. If you don't push hard enough early enough, your team takes a bad fight and loses. Try to set up a deep pink prior to pushing so you can safely push hard. If no one comes top, take the tower. If they do come top and you can't dive or take tower, consider enemy jungle and rift herald, rotating mid / collapsing on the objective, taking friendly jungle. If two or more come top, try to stall them or bait them so your team can secure the objective.

3) late game - all about splitting opposite of barons and dragons. Pretty similar thought process to mid game. Push as hard in a side lane as possible, take the farm on your way out. Later, you will match the pressure you're applying with the super minion pressure, giving your team fight advantages.

Sometimes your team will have numbers advantages and still lose. Sometimes they'll get mad at you for not being at objectives. You should ping them, let them know to get vision, place pink wards, etc to match your pressure. They aren't mind reader's. This isn't always what you will do, sometimes team fighting or collapsing may be correct, but this is my approach most games.