Without wanting to sound 'snippish', the Cliffs of Moher are IN County Clare, so specific advice would require more detailed information, as in, WHERE in Clare you are coming from.
I'm guessing that you don't plan on having a rental car, so either the bus,
As Bob has said Clare is a large county. He has posted the bus website for you. Another option is hiring a hackney cab. They are not on a meter and will tell you the cost of the trip in advance.
You also did not mention time of year you are traveling. There are day tours of the Cliffs of Moher from Limerick during the season. Let me know if you would like more info about them.