How to make carts go up an incline?

For a short term solution, a Powered Minecart is probably adequate if you're not going up severe inclines.

The extensive use of coal over long periods of time probably isn't worth it though. If you are going to want to continue to use this setup for a long time, you should look into powered rails. They're more expensive to create, but they also don't require any materials for continued usage. If you go with powered rails, you will be interested to read this previously asked question

The answer your looking for, other than a powered mine cart, is a powered rail.