Why doesn't my phone start after showing apple logo for 0.1 second?

When troubleshooting charging issues, I would do the following (in order):

1. Always try another Lightning cable and charger, preferably Apple original or MFi certified units.
2. The Lightning cable 8-pin connector should be sitting flush with the housing and completely inserted. If it is not, there could be lint/dust/debris inside the port impeding a proper connection. You can clean it out with fine point tweezers or a dental pick. Just be careful not to damage the pins inside the Lightning Port.
3. Change the battery. The battery is the weakest link in the entire phone and certainly for charger related issues. It is also the easiest thing to replace. Use a battery utility, such as coconutBattery (for Mac) or 3uTools (for Windows) to test the battery prior to removal. Anything less than 70% of design capacity will require replacement.
4. If the battery is in good condition, you can use a USB Ammeter to determine if the phone is really drawing current when it says it's charging. If it isn't drawing current, then I would change the Lightning/Charge Port (which you have already tried).
5. If a new battery and charge port still don't resolve the issue, then you either have faulty replacement parts (possible) or a logic board issue. There is an IC (commonly referred to as Tristar) responsible for communicating with the charging device and uncertified chargers can damage it. Unplug the battery and connect a known-good (preferably Apple original or MFi) charger to the phone. A properly working phone will show an Apple logo and boot loop. If the Tristar IC is bad then the phone will not do anything.