The secret to a really dry martini is very very little vermouth. 

You can use a martini shaker

Two jiggers (two shot glasses full) or the vodka or gin of your choice. 
I like Absolut Vodka (no flavor),
Alan likes Bombay Sapphire Gin.
About a quarter of a teaspoon of Noilly Pratt Vermouth (or less)
If you want to look like a pro – us the cap of the vermouth bottle to measure and eyeball it….

Fill the martini shaker with ice, pour in liquid.  Shake and pour into a martini glass.

If you don’t have a martini shaker, fill a large wine glass with ice, pour the liquids in the wine glass, stir with your finger and then hold the ice while you tip it and pour it into a martini glass (it works just as well depending on who’s watching). 

Garnish with either a twist of lemon or an olive depending on your preference.