Assuming you have already downloaded livesplit from the resources section of this page:

1. Right click the livesplit window and click on "edit splits".
2. Change the Game Name at the top to "Barji's Grappler!", and change the run category to whichever you are running. Section, or full game runs.
3. You should see a button that says "Activate", click this to turn on the autosplitter / autotimer.
4. Now just add your splits depending on your chosen category (e.g. full game run you should have 9 splits ready for each level).

Whenever you begin a level from the main menu on grappler now, the IGT timer will only go up if the timer in game is going up, and when you finish a level and press "continue", the split will be saved and the next one will be ready for splitting next!

NOTE: you will have to manually split the last split to save it.