Do you need a 'Li-ion battery charger circuit'? Then here's a simpler one designed by MrAl (wellknown member here), and drawn by me. It's nice but you need to disconnect the circuit after charging full. To know the status of full of charge, you need to measure battery charging voltage, if you got 4.2V then understand that the charging is full. Or you can mesure its charging current, it should be few mA flowing while full of charge, such as 15mA or 20mA.
Or if you need an indicator light to see 'full' then I can post another additional circuit with voltage monitor.