Usually Li-ion battery chargers are CL-CV power source. When the battery voltage is low, the battery will try to draw a lot of current from the charger. The current is limited to typically around 0.5C to 0.8C (1C = 1000mA for 1000mAh battery). In short, the charger ensures that the charging voltage is fixed to 4.20V per cell and may go lower when current limit is active at lower battery voltages. STC4054 is a simple single cell linear Li-ion battery charger with upto 800mA charge current. Advanced chargers may use coulomb counting and temperature monitoring for faster charging and battery SOC (State of Charge) and SOH (State of Health) implementation. If you can share more info regarding your end application and number of cells in series, it may be possible to share more specific feedback.
If you implement a constant voltage output block followed by a current limiter block or vice versa, it will do the needful depending on cell voltage and the charging current being drawn.
It is also possible to implement this in software using MCU with ADCs to measure output voltage, current and cell temperature along with a buck regulator controlled by Timer PWM signals.
