Check the Beckett website for complete details, but they now do ship direct back to Australia, so there is no need for you to have a contact in the US to grade with them.
First thing to do is either use the online submission form or print out the submission form from their website and fill it out with payment details etc. Be sure to include the amount for return shipping to Australia, which should be listed on their website or on the form (I haven't seen the form recently so im not sure)
What you need to do is pack your cards up with a copy of the submission form (preferably in card savers as per above posts) and tape the cardsavers in between two bits of cardboard (be careful not to put too much stress on the cardboard - just enough to make sure the cardsavers aren't going to fly around). Cardsavers are used because - well - that's what the grading companies want. They both say they accept no responsibility for cards sent in any other holders (BGS don't take responsibility full stop - but that's for another story). I then put them into another box and pad them with newspaper or similar material. Then pay for shipping to the US with Insurance - the amount of insurance is up to you. If the cards arent very expensive I will just insure for $100 - this is enough to give you a tracking number so you can see where your parcel is online.
Once received at BGS they will normally send you an email to confirm receipt. Then after the cards have been graded, they will send you another email to confirm they have been graded. Theyw ill then ship them back to you via Registered Mail (I know PSA does, cant confirm if BGS does) and you will get them delivered to you.
The old process where you had to pay for FedEX delivery back to Australia is long gone. Depending on the service level used, it can be done a lot more cheaper than you think.
After saying all that, I wouldnt use BGS if you paid me (search for other threads here). In my opinion PSA is the way to go.
Hope this helps you.