How to put a battery and battery charger parallel in a circuit
To be clear, this question is poorly worded, and you may find that to cause you a bit of trouble. Moving on, then.

Your charger output looks like it most likely is not designed for the battery you are using. You didn't exactly tell us much about the battery, or your application.

Assuming that 4.5 volts is not dangerous for the battery - and it might be - one of the neat things about dc is that load sharing is fairly automatic. Physics is on your side.

As long as the battery is in good health, and not going to explode because of the charger you put on it, the system will draw proportionally from the battery and the charger, depending on their actual terminal voltages and their actual output impedance. Typically, a well designed charger has a lower output impedance and a higher voltage than the battery it is attached to, and thus will assume most of the load being drawn.

A side effect of this, is that the charger will most likely put out as much as it can and get stuck so to speak. The battery will not only fail to charge, but likely will continue discharging to power the load.

This is where you come in. You have to design the system such that the battery will charge when its supposed to, and still maintain continuity of power to the load. Proper battery chargers with current monitoring functions and state of charge estimation are great little projects to learn on.