What is the best way to level up One-Handed in Skyrim?

One-Handed XP is gained when you deal damage to valid targets, and thus you should attempt to deal as much damage as you can in a short amount of time to a valid target.

Having a better weapon helps, and you can also whack on things that are marked as Essential and/or have very high health. The One-Handed page on UESP notes that using a grindstone to improve your weapon or boosting your damage with spells doesn't increase XP earned.

Since you've cleared the Dark Brotherhood quest line, Shadowmere makes a good punching bag. Shadowmere isn't completely invulnerable, but is damn near close, since he regenerates health super fast and will respawn after 10 days if killed. Pick a fast weapon and get to slashin'

Alternatively, you can go visit a trainer. Athis is the Expert One-Handed trainer, can train you to level 75. As he's also a follower when you complete the Companions quest line, you could get your gold back via trading with him if you so desire.

Beyond that there are a few small bonuses from reading books and some odds and ends quests, these are all given on the UESP page for One-Handed.

I know its too late to help you after completing so much, but... during the very first mission, when you either follow the guard or the stormcloak, choose the guard.. eventually you come up to a bear and he tells you to lead. Instead of leading, just hack the crap out of him. He doesn't turn on you and regenerates health, so you can farm XP.

I literally JUST found this out with a new orc. My One handed is up to 50 now.. ive been slashing away while I wrote this.. gonna move on to 2 handed then destruction magic haha