The TP4056 does all of that automatically. The datasheet shows that it monitors charging current, and when it drops below a certain level at which the battery is almost fully charged, it terminates charging. It won't resume charging until battery voltage drops below a certain level. So it doesn't trickle charge the battery all the time. It actually shuts down the charging completely, and doesn't resume unless the battery has been discharged a bit. It's unlikely an Arduino could add anything useful to this process.
However, since charge termination is based on the current draw detected by the TP4056, you need to consider whether your circuit will be powered up while you are charging the battery. If it will be powered, then current drawn by the circuit may confuse the TP4056 into thinking the battery is still charging, and it may never shut off. That could be dangerous. That can be solved by having your circuit go to sleep from time to time so it draws very little current, which allows the charger to terminate charging properly. Or you can add a "load sharing" circuit consisting of a mosfet, a Schottky diode and a resistor which lets the incoming power source directly power the circuit when it's plugged in, so the charger won't be confused. And of course if your circuit draws less current than the shutoff level even when active, then you don't have to worry about termination at all.