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Here is a link to the online RC parish register for Kilfarboy RC church. http://registers.nli.ie/parishes/0788 Baptismal records start in 1831 and marriages in 1856. Civil registration did not start until 1864. Anthony and siblings would have been baptized before 1831 and I assume his marriage occurred before 1856 so you may not find any info on them but possibly you can find baptismal records for their children.
The 1855 Griffiths Valuation head of household listing for Kilfarboy civil parish does show Anthony and two other Cleary records in Illaun. http://www.failteromhat.com/griffiths/clare/kilfarboy.htm
You may also want to check the 1901 and 1911 censuses for Illaun.