My portable charger wont stay on. How do you make it stay on to charge the phone?

If you're using the charger supplied with the phone, it should either work or not. If it doesn't; either the phone battery, charger or both require replacement.

If the battery is "good" and you are using a "3rd party" (non-original equipment) charger, you may not have one that is incapable of providing the correct voltage, charge current or both. If the charger can not provide adequate charge current or voltage, it is very possible that the charger is shutting off due to excessive heat. Once sufficiently cooled, it charges again.

Make sure that you've got the correct charger - or one that is designed to charge the type battery in your phone at the correct voltage and current rates.