The general rule of thumb when pruning is not to remove more than 1/3 of the plant material in a single growing season. So have patience, this may be a multi-season project. I have a lot of experience pruning trees and have seen some nightmare hack jobs. As with many fruiting and flowering trees, a little pruning each year may be all it needs to improve its growth habit. But now is the time to prune it, before it leafs out fully and begins to set fruit.
I live in Lansing, MI. If you live close, I may be able to come show you how to prune this tree. If not, you could post a photo or two.
Otherwise, here is some general advice. Start by removing anything that is DEAD, DAMAGED, or DISEASED. Once that is done, if there is still more that can be done this year (up to the 1/3 rule), remove branches that rub against other branches, criss-cross, or grow awkwardly. This includes the straight, upright "water sprouts" that are so comon on apples. Also, remember that you may not have to remove an entire branch. You may only need to cut it back to a joint or a V. You can control the directionality of a branch by doing this as well. Example: If a large branch is growing outward, and it splits off in two directions, one part of it might be growing over your yard and the other over your neighbor's. You can remove the unwanted part over your neighbor's yard while encouraging the more desireable part over your yard to grow. This concept works for large and small branches, and can be used to shorten overly long branches too. Apple trees tend to have what we call "twin" or "double" branches aswell. Which is just two branches that seem to mirror eachother and compete for the same space. One of these should be removed from each pair. Just choose which is more disirable relative to the rest of the tree. With apple trees, the goal is to have an openness to the growth pattern, which allows for better light penetration and air circulation around the fruit. Branches should also be encouraged (thru pruning) to grow up and out. Many times, apple trees will send out a branch that grows slightly downward or back in toward the center of the tree. These should be removed. Always make your cuts at natural-looking points such as a joint or fork in the branch. Also, start small, removing only small amounts at a time. You can always cut more off, but once it's cut, you can't re-attach it.
Absolutely do NOT dig it up to add soil! The root disturbance would likely killl the tree. You might try staking the tree to slowly pull it to vertical. This takes a while though (like years, depending on severity). You must also be careful to reposition the tie from the stake at least once per season to prevent it from becoming ingrained into the bark of the tree. Also, look around, is there something blocking sunlight from this apple tree? Another tree? A building or fence? All plants seek to grow toward sunlight. If something is blocking it, that could be the cause of the leaning. Allowing full direct sunlight to the tree may help encourage more uniform growth. The pruning may also help to lighten it's load so it won't need to lean on the fence for support.
For more information you can check out you local college or university's extension website and look for pruning tips. The Michigan State University Extension's website has lots of helpful information on many lawn and garden questions. You should look for photos/digrams of how to make proper pruning cuts. Bad cuts can be harmful to the tree, and they look terrible.
I hope this helps. Good luck!