Try swapping in a known good battery and charging port if you can get access to one. Then try plugging it into iTunes, if it turns on perfectly fine, then narrow it down to one part. If it doesn't connect to iTunes, your board is most likely bad.
Try to find a charger with a high amperage. Sometimes Android phoned need to get some serious juice pumped into them before they will power up. If you can't find one, look for a hard wired Micro USB phone charger (non-detachable cable). This issue has happened to me several times with various phones and it has worked. Also, leave it on the charger for a while.