Cold reduces a motorcycle battery's capacity by 30-40%, so it's normal for it to be weaker in winter. But if it's completely dead after two weeks without riding, there's a current leak somewhere - either the battery itself is shot, or something's drawing power even when the engine is off (alarm, GPS, stuck relay, etc).
Before you test the alternator, first check if there's a parasitic drain. Disconnect the negative cable from the battery and put your multimeter on amps between the cable and the terminal. If there's current flowing (more than a few milliamps), you've got a leak. It could be something silly like a light staying on or faulty electronics. That's the thing a lot of people forget to check before blaming the alternator.
If there's no parasitic drain, then you can test the alternator: engine running, measure the voltage at the battery terminals with your multimeter. It should read 13-14V. If it's lower (like 12V or less), the alternator isn't charging enough. You can also do a simple test: if the battery lasts several weeks normally but only dies in winter, it's clearly a weak battery plus cold problem, not the alternator. Buy a new battery for 50-60 bucks and test it - it's cheaper than an alternator anyway.