Couldn't be easier.
You just hire a plane (I mean "a private plane charter") to fly you there, and another one to fly you back from the other side.
Any jet charter in NZ will do it, and any number of companies do it enough they actually have Google-Ad-Words web sites for the function.
Simply google "private jet charter, NZ to Antartica".
Or phone any major jet charter company in NZ (or perhaps Tasmania, or just Melbourne - I mean what the hell is the odd thousand miles one way or the other here?) and ask for a price for the two legs.
My guess is it would actually cost less to hire a jet than to have a boat/ship of some type take you on both legs.
I'd guess the cost of two such jet rentals would be perhaps around $150,000 all-up.
(The tiny figures mentioned elsewhere, 20 grand etc, are almost certainly for a tour of some 10-30 I'd guess people, and it seems OP does not want a tour-bus approach - and fair enough!)
It's very easy to get to Antartica, you just rent a jet - in answer to your question.
Has anyone ever crossed Antartica solo ?
It looks like yes, a couple different blokes have arguably done this:
It takes two months.
Interesting, there is a HUGE difference and debate between unsupported and unassisted in the Exploration Thrills world.
Has anyone actually "crossed" Antartica anyway ?
NOTE that these dudes took the shortest imaginable route "across" (ROFL) Antartica:
OP, the field is wide open to be the first to actually go "across" Antartica.
Actually it looks like a Norwegian bloke, Borge Ousland, actually did it properly without using roads back in the 90s.
Random photo of impossibly rugged Norwegian Borge Ousland who did what OP asks about.
Seems he ingeniously used a kite where possible, like ski-kiteing.