If I'm asking it is because I don't know in which context I should use it, I think it is correct to say "My leg aches" and "My leg hurts" (I don't know if there is any difference here), but also I think it is correct "I hurt my leg" while "I ache my leg" is wrong. So what is the difference in those three correct options?
Hurt and ache can have very similar meaning and can sometimes be used interchangeably. I will try and give you some examples.
Both hurt and ache convey the idea of sensing pain (dolor). If you cut your finger, you could say either - My finger hurtsor My finger aches. The meaning here is almost identical. The verb in both cases is used intransitively, and that is the key point. Ache when used as a verb is used ONLY as an intransitive verb, meaning it has no object. As far as I know it not used as a transitive verb, but hurt is used in that way. You might hurt someone by pinching them, but you would not ache someone. That is the main difference. You said you were interested in the verb, but you might want to know something about the term ache when it is not used as a verb. In the US we often say both my stomach hurts or my head hurts, it is less common to say my head aches or my stomach aches. I think it is most common for an American English speaker to say I have a headache or I have a stomachache in preference to the other examples. This would also be true of a tooth, but I can't think of it being used with other parts of the body, such as hand, foot, leg etc. You would not say I have a footache or an armache. You could use the term heartache, but it would not refer to actual physical pain but rather an emotional or mental distress.
So to sum it up, I think using hurt as a verb is more common... you can't go wrong with it, so when in doubt, opt for hurt as a verb. Don't use ache as a transitive verb. My dictionary classifies it solely as an intransitive verb. The expressions to have a headache, a stomachache or a toothache are used, in my opinion, much more frequently when describing pain in those specific body parts. For other pain I would say hurt although you could say ache and be perfectly correct. I think ache is used most commonly when referring to emotional pain - such as my heart aches for those hungry children. I suppose one could say my heart hurts for those hungry children, but I think its use is very awkward sounding and less desirable.
"Hurt" can mean to cause pain or to ache, and in conversation/general complaints it is much more common to use "hurt". "My leg hurts" is much more common than "my leg aches". However, with certain body parts there are exceptions. Saying "my head aches" (or in a romantic sense - "my heart aches") doesn't sound so weird, although you can also say "my head/heart hurts" and still convey the same thing - perhaps in a less formal way.
To my ear, and this may be no more than personal linguistic baggage, 'my leg aches' carries overtones of a pain that is more dull than sharp, more longterm than sharp or sudden; whereas hurt (in the intransitive sense) suggests a more severe pain.
I forgot about backache as another term used frequently like headache or toothache. You may want to view the BBC website to get a very detailed explanation of hurt and ache, in addition to pain, sick, and ill.