It's great to hear you will be in Ireland this summer!
You can see your options on Rome2Rio
. It's a great site for this type of request.
From what I see, your best option is to take a bus from the neighborhood of Drumcondra in Dublin to Chapel Street Slane. You'll then have to walk 20 minutes from Chapel Street to Slane Castle. The journey should last 1 hour and a half.
Je vous souhaite un excellent voyage !