How do you open the Dark Palace without paying Kiki?

The first dungeon in the Dark World is the Dark Palace, as you approach the Palace a little monkey-type creature called Kiki starts following you, if you give him 10 rupees. Once you get to the entrance to the Dark Palace he offers to open it for you for 100 rupees.

You are also given the opportunity during this conversation to "try and do it yourself". Is it possible to open the Dark Palace without paying Kiki rupees, if so - how would you achieve this?

No, there is no other way of entering the palace without paying Kiki:

Kiki will agree to accompany Link to the Palace of Darkness for a payment of ten Rupees; this must be paid, as Link cannot enter the palace without Kiki's help. When they reach the dungeon entrance, Kiki will open it for Link in exchange for 100 Rupees.

Source here

While I'm sure it's not what the script writer intended, this is regularly done in speed runs by exploiting a glitch in the game. It's extremely difficult to do and requires frame perfect input.

I have completed A Link To The Past and the only way to get in is to pay Kiki. The best way to get rupees is to cut tall grass, that gives out blue or red ones.