Your phone is corrupted. You will have to factory reset the immediately let it update, back up your data first Update (10/13/2016): My Blue studio xl was exactly same way. Sometimes the updates can cause phone to "bootloop" if you interrupt any part of the update. I got impatient after installation because BLU logo screen stayed on for a few minutes but it was part of the process.
Have you tried to charge using a PC or laptop. I had a phone that would only charge on PC until I flashed phone Update (10/12/2016): Please reply for more help
You will need to flash your phone, on a scale of 1-10 it is about a 3 to do. Let me know if you want to do it. All you need is a computer an a program Update (10/12/2016): First try putting your battery in, next as you press power button press volume down at same time to get to boot menu, go to factory data reset and hit OK. After complete go do wipe cache partition, then restart.
You need a computer, firmware, and MTK Flashtools and about 30 min. I can help if you like. Download firmware from here https://www.mediafire.com/folder/y96u64b... Download SP Flashtools and MTK DRIVERS from here https://www.mediafire.com/folder/e3r6ens...