You don't need a larger alternator to handle a laptop computer connected to the cigarette lighter. A laptop would only draw a few amps.
It is most likely coincidental that you experienced electrical problems with the connection of the laptop unless there is a short in the laptop power cord or charger.
The alternator simply recharges the battery when the engine is running. To check your alternator, simply start the engine and put a DC Voltmeter across the battery terminals. You should read around 13.5 volts. If you're reading only about 12 volts, the alternator is not charging.