Hi Cameron,
June is a bit warm, but it's preferable to walking in July and August when it's a lot hotter and crowded. So try to start early June.
Getting to SJPP? Well it probably depends where you live in Australia. Here are couple of ways from the east coast, both with Qantas as that's the way I've done it:
1. Fly from Sydney to Paris, then get the fast train (TGV) from Paris to Bayonne, then local train from Bayonne to St Jean.
http://www.voyages-sncf.com the French rail website will give you more info, just scroll down the page and click on the Union Jack. There are other airlines that do the Sydney to Paris run.
2. Fly Qantas from Sydney to Madrid via Frankfurt, get the bus from Madrid city to Pamplona, hang around there for a couple of hours, get the 6pm bus to Roncesvalles, and taxi from there to SJPP. The taxi will be waiting for pilgrims at the Roncesvalles stop.
Another alternative is to fly into Heathrow, then get to one of the other airports and fly to Biarritz. But I've no idea of the logistics of this.
Buen Camino.