You must learn about the dangers of charging and discharging a Lithium-Ion battery to avoid an explosion and/or very hot fire. Usually a charger IC is used that is programmed for the spec's of your battery. The IC has many safety features in it.
The phone already has a charger circuit in it. It is powered by a 5V power supply and the battery cell that you will use must drive a boost converter to boost its voltage up to 5V. Your "custom portable charger" is commonly available as a Power Bank.