Is there anyway to get free music on iPhone 5c?

With paying for college, cosmetology school, & rent & bills. It's just out of my price range to purchase something such as music.

You could download YouTube or Vevo.
They are both free, but if you want to listen to music without internet there are apps where you can download audio or the actual video off YouTube or other video streaming sites.
Some of them you have to pay for and you can download unlimited audio and videos and others let you download a certain amount of videos until it asks if you want to upgrade to an unlimited amount of videos.
Apps are usually 69p (In UK currency) which shouldn't make too much of a dent in your budget.