There are many versions of an apple martini (or appletini). Some use calvados, some apple puree, and many of the newest use apple schnapps.
This recipe is the easiest by far, simply mixing apple schnapps with vodka and it has a very sweet taste. I did add a touch of lemon juice to this one because it balances out the sweetness of the schnapps.If you'd like to have a Sour Appletini, use Sour Apple Pucker. In that case I would skip the lemon completely.
The color of the drink is going to depend on the brand of schnapps you choose as some are greener than others.
For a garnish, you might go with a green apple slice or peel that has been dipped in lemon juice to preserve its color.