If you have a simple volt-ohmmeter, you can do the following as initial checks.
The numbers below may vary from car to car, so I have indicated them as ranges.
With the engine running, measure the voltage at the battery terminals. If the charging circuit is working, the meter should read more than 12.6 V and should increase with more engine speed to a max of about 14.2 to 14.5V. If it's much less than that or greater than that, there is a problem with either the alternator or the voltage regulator.
With the engine running, measure the voltage from the alternator output post (B+ post) to the battery positive terminal. This should be 0.3V or less. Higher readings indicate too much resistance somewhere in the circuit or an alternator that is trying too hard to generate amps.
There are ways to test more, but require more equipment.
As I said in my previous post, I would check the battery first to make sure that it's in good condition!
I sorta disagree with this statement: "the battery is not used for running the car in anyway".
The electrical system in any car is usually a balanced affair designed by the manufacturer to give satisfactory performance under average driving conditions with the installed factory accessories. If all these parameters are met than you shouldn't have to change things such as, alternator pulley diameters, battery size, alternator size(output).
I don't like to "jump the gun". I like to do these initial checks first. ;)