I am aiming for the "kill legendary dragon" achievement. I need to be level 78. I'm level 76. I've grinded everything else to level 100. My alchemy is 22 my enchanting is 32. I've leveled these up naturally. Not sure how to successfully powerlevel them.

Try this link. It might help.

http://www.gamesas.com/topic/1481759-best-alchemy-ingredient-combinations/

Good list, I will try this out.

for alchemy, salmon roe, salt pile and hist carp.

A basic but effective grind for Alchemy is using Blue Mountain Flower and Blue Butterfly Wing.

Won't it be hard to level up, just by leveling Alchemy and Enchanting? If so, you can install the http://www.nexusmods.com/skyrim/mods/1175/?.

Yup. It will be very hard. But everything else is maxed out. I'm almost 77 anyway.

Why not turn some other skills legendary?

Yes indeed, probably the best idea.