How to beat the boss in level 6-3, Final Trial?

I had the same problem; after you shoot the gem four times and the attack pattern changes, it seems you can't hit the gem any more but you can; you just have to be annoyingly precise about it.

Here's what I did: When the level starts run right up to the boss and wait for it to jump over you. A block will fall behind the boss, use it to jump on its back, shoot the gem, and get off. Stand next to the boss again and wait for it to jump backwards over you. After it lands move just far enough forward to avoid the boss fire attack. This screenshot shows a safe place to stand.

When the fire attack is done, walk back up to the boss and let it jump over you again. Again, shoot the gem. Do this two more times to stop the boss breathing fire.

Now it will just jump around raising the spikes on its back and causing more blocks to fall. You still just have to use the block that falls behind it to jump on its back as soon as the spikes on its back retract. I like to stand in the safe zone on the far left, run under when it jumps, and use the block behind it to get on.

You have a tiny window of time to shoot the gem and the boss will probably jump as soon as you get on its back. It took me quite some time to pull it off, but you only have to hit it once more and the fight is nearly over. Stay on its back; do not jump off!

Finally, after the fifth shot, the boss will stop and jump straight up. More blocks will fall and spikes will come out of the ground. If you stayed on the its back you just need to dodge any falling blocks. If you jumped off you can use the blocks to get back on and avoid the spikes.

Good luck! Even knowing how to do it, It still takes me several lives to get that fifth shot to hit. Perhaps there's a better way to do it, but this method worked for me.