To be honest, these days the risk and hassle of driving in Central London, let alone trying to find somewhere suitable to park, just isn't worth it. Last time I did that about 13 years ago, I parked under Hyde Park overnight and it cost well over £30 even for a Saturday night. The trudge to the Club was a bit of a pain too.
When I was still in the RAF, I used to drive to RAF Northolt, leave the car at the OM car park, then walk to Ruislip Gardens Underground station and take the Central Line to Oxford Circus, then the Victoria Line to Green Park. Which takes about 1:20 if you're lucky.
If you can't park at RAF Northolt, oddly enough another alternative is to use the London Airport Long Term car park (some good weekend deals), free shuttle bus to Terminal 2, then the Piccadilly Line to Green Park. It's easier than Hyde Park Corner as you don't have the hassle of working out which beggar-inhabited subway to use or to fight your way past hordes of Orientals with their infernal phones on sticks outside the Hard Rock. But if you've got a lot of luggage, it's uphill from Green Park to the Club and you'll certainly notice it.
Nowadays I'm very fortunate to have Oxford Parkway not far from here. There's usually plenty of parking space (£8 for up to 2 days), trains every 30 min to Marylebone, then the Bakerloo Line to Oxford Circus and Victoria Line to Green Park.
Oh for the days when I used to whizz up to Town in my MG Midget and park in a car park which had a discount deal with the RAF Club!