You don't need to be a US citizen to buy firearms, but you have to be "legal".
1. when you enter US you will be given a worthless looking piece of paper with an alien number;
2. after you enter the country apply for a Social Security Number (SSN) and get a driver's license
3. save the rent, electric, garbage, phone, and other utility bills for 3 months.
Buy a hunting permit, and take 1, 2, and 3 with you and buy yourself the firearms that you want from the many stores that sell them.
If you have some particularly valuable or heirloom guns, you can import those ino US, but it's a bit more complicated and costly. It will take import paperwork and fees. Better have them sent by mail, don't show up at the border with guns. This is worth doing only for special guns. If your guns are replaceable, you would be better off selling in Australia and buying here in US.
Forget about Remchesters, get a Tikka.
A bit late, but i hope that it helps. I just talked on the phone with some "federales" to make sure the info is accurate.
One last word of caution: if you are associated with Pakistan, Yemen, Iran, and other such nice places, don't try to get any guns in US.