Let's start with Vodka. What a fucking waste. Vodka, at it's best, is supposed to be flavourless. So you're already off to a terrible start. If you're asking for a Vodka Martini you're asking for Strong Vermouth.
So let's get that out of the way. You need the standard gin. Gin is delicious. There are many different Gins to choose from. Gin has actual flavor. I like Hendricks, but try and figure out which Gun you like best. Beefeater is great for a gin & tonic, might not work as well in a Martini. Try a few different options.
Now we've got Gin and vermouth. Those two work together beautifully.
Now I like a good olive. I say make that fucker dirty. Extra dirty. OLIVE ME THE FUCK UP! That's what I want. You want it straight up? Fine. I think it's packing but it's the traditional way and I won't complain. You want a cocktail onion? That's called a Gibson and I ain't complaining.
But GIN THAT FUCKER UP. Fuck off with your weak ass Vodka Martini. I don't care if it's Grey Goose or some other "fancy" shit. That ain't no Martini.
And make sure to use enough vermouth! None of this “wave the vermouth bottle near the shaker” nonsense. A martini should be about 1/4 vermouth as a starting point.
I remember hearing a theory that he deliberately orders a mostly flavourless cocktail with very basic and common ingredients because it would make it easier to detect if someone had spiked his drink with something.
Standard, off the shelf ingredients means you can't just spike the whole bottle ahead of time as each ingredient is pretty standard.