How to backup files from black screened iPhone
Check any computer you connect the phone to see if it has a backup. Then check in iCloud.com to see your last backup. If you have paired a bluetooth keyboard, use that to unlock the device. Also - the screen could be out but the digitizer might still work. Can you put another iPhone 6 next to yours and go through the motions to enter the PIN by tapping where you know the touch areas are on the screen?

At that point, you'll need to make a decision based on cost and time you can be without the phone while it's repaired or the data recovered. 

The data on the iPhone is almost certainly encrypted so you'll need to pay someone to make a forensic-type repair on the phone to replace the screen without damaging the storage. That costs more than a repair where you don't need to keep the data intact. Also, you can't pair the phone to a new computer without pairing a bluetooth keyboard or repairing the digitizer / screen.

Once you have quotes, you can make a decision based on the availability of reliable service where you live. If you took your phone to a skilled technician at the Huaqiangbei electronics market in Shenzhen - you might pay $50 for the repair and have your pick of 100 technicians in a 4 block radius. Other places, you might have to pay $5000 and ship your phone.

Paying someone to decrypt the contents is probably a factor of 10 more expensive on both the low end and high end unless you could tell them your passcode.

So:


spend half an hour assessing the availability of backups (and maybe time to go visit computers you could have synced to)
make a buy a new phone decision if you are going to shop around or ship your phone for repair to the cheapest shop you trust to make a custom repair. (you can then sell either the repaired phone or the new phone based on whichever recoups your investment in spare hardware to allow a cheaper repair).