First, set up your splits. There are plenty of Livesplit guides on Youtube that explain it much better than I could, so go to one of those if you don't know how it works already. Then, either download the splits for Level 15 or create your own.

Once you're in the game, get your bearings and make sure you're totally loaded in before you move. Then, start the timer and start running. If you can, you can use the mushrooms to make it to town more quickly. However, if you fail you can lose quite a bit of time if you are flung away from the town. With mushrooms or without mushrooms, keep running until you reach the Mushtown shopkeeper. Ignore or jump over them, and go behind them into a hole. There will be a bluesteel chest here. Open it, and keep running. You want to head to the area with the baby shrooms, which is also right behind the shopkeep. The guard will try to talk to you, so be ready to exit out of that conversation quickly. Then, take a left and start killing baby Mushrooms. Level up, grab double slash, and then level up again.

Then, while running, allocate your stat points (Put at least 4 into DEX preferably).  Backtrack to behind your spawn, so that you can enter the Great Crossroads.  From there, equip the wooden sword that you got from the bluesteel chest, and use the Scarecrow cheese to kill a few.  This should give you a good bump in levels.  Allocate your stat points, then do the Brutus Skip to enter Nilgarf.  From there, you want to jump to the premium shop and run through the pond.  This will take you to the Sewers.  Here, you have access to plenty of walls and grids to cheese Batties, so you can farm to level 15 extremely quickly.