How can I get my music off my phone when it's DEAD?

I have an iPhone 6. I don't have phone service, so I just use it for music, internet, emails, and social media. Yesterday, when I woke up, the phone had the symbol on it for a dead battery. It is still like that even though it has been charged up since. I have been having a lot of issues with it where it keeps dying on me when it still says I have enough percentage, plus I only have about 20-30 minutes usage before it dies. I got the phone when I was getting student loans in college. I have since graduated and have been having trouble finding employment, so replacing the phone is out of the question.

My problem is, I logged onto iTunes and I found out that the music that I have purchased, I didn't download it to my computer. I usually download my music on my phone so I never thought to download it on my computer. My question is, is there a way I can get my music back on my computer without using my phone since my phone is dead?

You can't

Just replace the battery.

Well since the phone is dead, you can't really retrieve any data off of it.
However, you mentioned that the music you have you purchased it on iTunes on your phone, right? If so, you can simply sign in on iTunes with that account that you used to purchase the music and it should show up in your purchases on your iTunes on your computer.

You have to get the phone working first…

Try a reset - both home and power buttons at the same time, for 10 - 12 seconds, until the apple logo appears.
Do that after it's been on charge for some time and with external power still connected.

If it works, then connect to a computer and run "syncios" - you can use that to copy files from the phone to the computer.

(Go to the "freeware" tab for the basic copy tool).

I'd expect that anything bought via your apple account should be available on your computer within itunes, or on any device on that apple account - but syncios can also copy files that were not bought via itunes as well as those that were.

The same way you would if you were trying to listen to a CD on a broken CD player. You don't.

Step 1: Repair (not replace) the broken playback device.

Hindsight being 20/20: When the phone was working, backing it up to a computer would have provided the music files you want.

You can sing them into your brain memory

You are between a speeding semi truck and a granite cliff.

Have the battery in the phone replaced.

You can't

You can't

Music purchased normally form the iTunes store will let you download the music to any device for free. Otherwise you can just take your old iPhone to any Apple store or "strip mall" cell phone repair shop who will replace your battery for $30-$50.