First of all you must join the ARA and particularly RAINF. If you are not an infantry soldier the only way you could possibly become a Sniper is to conduct a Corps transfer or apply for special forces.
If you are RAINF, you would know about the procedures already, so I'm assuming you aren't.
Once you complete basic, advanced basic and IET you will be posted to one of the high readiness infantry battalions as a rifleman. Once you've served a period of time as a rifleman (usually 12 months or greater) you will have an opportunity to be posted to a "specialist" sub-section of the battalion, specifically mortar platoon, direct fire support weapons platoon, assault pioneer platoon or if you are good enoug, "recon" platoon.
Within the Recon platoon of our regular battalions (and some reserve battalions) is a section known as the Sniper Cell. Anyone within a battalion can apply for the Sniper Course (which is run at Singleton) but realistically unless you've done a posting or 2 in Recon platoon, you're unlikely to have the fieldcraft necessary to successfully complete the Sniper Course.
If you manage to get on the Course and pass, you will be posted to the Recon platoon of your battalion and placed in the Sniper Cell as a number:2 with an experienced Sniper, (usually at least a Corporal) once you have done this role for a while you can undertake further training to become a number:1 and actually be a "Sniper".
Remember though it takes a long time and plenty of dedication. It's about the hardest job in an infantry battalion and only the very best are chosen. There are only about 60 positions for Snipers in the entire Army, so as you can see it is often extremely difficult to get in...