One handed can take a long time to level up in the ordinary fashion. However, if you use the techniques I have outlined below you can solely focus on leveling one handed, speeding up the process significantly.
As with destruction you will want to conjure a creature using the conjugation magic. It is best if you cast the spell that will create the creature with the most health as one handed levels up according to how much damage you deal to a valid opponent. make sure you have some restore magicka potions ready so that you can continues this techniques for a while because on the later levels it takes quite a lot to level up this skill.
The best trainer is Chief Burguk who can be found in Dushnokh Val which you can locate on the image below. Remember that the gold you paid for training can be pickpocketed back as long as it isn’t too much gold (around 2000 gold the chance of pickpocketing approaches 0%).