Like Mike said, you can get some of your requirements but not all together. Trips departing from New Zealand and Australia are 3-4 weeks and start from about $15K-$20K. It's like that because of the distances involved and the liklihood of encountering heavy ice.
If you go for a Peninsula trip though, you should be able to stick within your budget, visit a scientific station, cross the circle and possibly spend a night or two under canvas. Whatever trip you take it'll be one to remember forever though. Helicopter trips unlikely as it's usually only the ice-breakers that carry those and they are used on the Australian / NZ side of the continent - which is where this started.
There are some trips started recently where you can fly/camp to Antarctica from Chile. Once you're there though you don't get to travel about so much as you would on a ship.