Hi, a couple of suggestions for you.
1. As you are in Australia, you can join the National Library of Australia, and with your library card, you can access the Ireland newsarchives. Normally this is a subscription site, but access is free via NLA on-line resources. Unfortunately, it doesn't have the Clare newspapers, but references to Clare people do come up in papers from surrounding areas.
2. NLA hold Clare Journal on microfilm, which you can borrow via inter-library loan through your local library. You would need access to a microfilm reader (I had to go to local university library for this), and it is a tediously slow process. Would be quicker if you knew approximate date range to narrow your search.
3. Limerick city have online obituaries from 1831 onwards from the Limerick Chronicle. Unfortunately, they only index the ones actually from Limerick, so you would need to manually scroll through to find any of the many Clare entries. These are at:
http://www.limerickcity.ie/Library/LocalStudies/ObituariesdeathnoticesetcfromtheLimerickChronicle/Hope this is of some help to you.