My leg hurts.

This is a very general way to describe pain in your leg. There could be many different reasons why your leg hurts. Maybe you cut it or bruised it. Maybe you pulled a muscle doing some exercise. Maybe it’s cold outside and your bones can’t take it. Maybe you’re not really sure why it hurts.

My leg is sore.

The best way to use this sentence is if your leg hurts because you exercised too hard. Maybe you went to the gym yesterday and gave your muscles a good workout. So today your body is sore (your muscles are sore).

I hurt my leg.
I injured my leg.

These two general sentences mean pretty much the same thing. Something more serious happened to your leg. If you just said one of these sentences, the person you are talking to would probably respond by asking ‘how’ you hurt or injured your leg:

How did you hurt your leg?
How did you injure your leg?

Or they might ask you ‘what’ you did to your leg:

What did you do?
What happened?

You can give more specific information about your leg like this:

I hurt my leg playing basketball.
I fell down the stairs and injured my leg.
I broke my leg in a car accident.
I fell off my bike and cut my leg.
I bruised my leg playing football.