You need to check for the availability of (at least) 3 things - 1) business name (ASIC), 2) domain name (use WHOIS/domain name lookup on a site such as webcity.com.au) and 3) trade mark (definitely a specialist lawyer, probably a patent specialist).

The first two are simple, cheap and you can do them yourself. The third will probably be moderately expensive, and must be done by a suitably qualified and experienced lawyer - it's a real minefield ... !