As I understand it, Li-Ion charging is normally in two phases. The first is the constant current phase where the current is limited, and then the constant voltage where the voltage is limited to 4.2 V.
In the constant current phase, the current is limited to what the battery can take, or what the charger will take. It doesn't much matter if the current isn't constant, as long as it doesn't overheat the charger or is too much for the battery. What batteries will take varies, but most will be happy charging them at the 1C rate. If your current isn't constant, charging could take longer, but that is the only problem.
Your 4.2 V has to be very accurate, and you might get a larger number of charges at 4.1 V.
There are a couple of other rules that commercial chargers apply when needed. If the battery voltage is low, below about 3.2 V, charging should not be more than 0.1 C. The characteristics of the batteries mean that it takes very little time to come from 3 V to 3.2 V even at 0.1 C.
Also, you should not charge if the battery temperature is below 5 or above 40 C