How do I determine the alcohol content of a homemade drink?

I'm not sure about Brandy, but in beer and winemaking, what you need a simple tool called a hydrometer. It measures the "specific gravity" of any liquid. You don't really need to know what that means, but in very basic terms it's the density of the liquid.

The first measurement is the Original Gravity (OG). That's the one you measure right before fermentation. The second measurement is the Final Gravity (FG). You do that after the yeast is done fermenting. It will be lower, because the alcohol will be there instead of the sugars that the yeast gobbled up. You take the two numbers and do some simple math to get your Alcohol by Volume (ABV): (OG-FG) x 131 = ABV

You can get a hydrometer at any local homebrew supply shop, or order one online at a place like Northern Brewer. As I looked at Northern Brewer, I saw that they have a "Proof and Tralle Hydrometer", which is used to measure alcohol in distilled spirits, but is "useless for beer or winemaking". At that point we're out of my area of knowledge. It would definitely be worth calling them up and asking.

Thanks BW! Actually, I think it is the Proof and Tralle Hydrometer I need, since this drink consists of whiskey added to a whole bunch of sugar and berries.

Thanks for the help. I'm checking the Northern Brewer site now.

Awesome. Glad I could help. Definitely give the northern brewer people a call. I believe they're very helpful folks.