Make the screen always be off when having a call
Sounds to me like a bad ambient light sensor and/or the proximity sensor. Both are a part of the front facing camera assembly, and you cannot buy them separately. 

It should be covered under warranty, but if it's a result of a drop or water damage, then it's not covered under the standard warranty and you may have to have it repaired, either by yourself or a pro.

If you feel comfortable with the repair, here are some tips:


When removing the digitizer assembly (touchscreen) with a suction cup, take extra care to avoid severing the digitizer and LCD cables.
If you damage the home button or its cable, then TouchID will no longer work.
Here's the guide


Of course, you could always take it to a pro, just be prepared to pay likely $100+, unfortunately. Maybe even $150+, depending on where you go.

If it's not under warranty, though, wait until the iPhone 6S comes out in less than a couple of weeks, as once the iPhone 6 is no longer the newest, parts and repair costs should be lower.