How to determine the alcohol content of a mixed-drink?

To a first approximation, you can just calculate the volume of pure ethanol in each of the alcoholic ingredients (total volume * %ABV = volume of pure ethanol), add those up, and divide by the total volume of all ingredients.

For example, a margarita with 35mL 40% ABV tequila, 20mL 40% ABV Cointreau, and 15mL lime juice would have alcohol content of (35mL*40% + 20mL*40% + 15mL*0%)/(35mL+20mL+15mL) = 31% ABV.

That's an approximation because the volumes are not strictly additive: mixing 50 ml pure ethanol with 50 ml water will actually give you something like 95 ml of the mixture. To get a truly accurate answer, you'll need to measure the volume of the final drink, and divide your total alcohol content by that.

If there are no significant other (non water, non alcohol) liquids involved (especially not heavy amounts of sugar), an areometer/spiritometer (available as a brewer's tool,$10-$20) should be able to tell you.