Dublin airport to Heuston station

The "best" way is by private car or taxi.

If you are looking for something less expensive, the Dublin Express bus #782 goes from the Airport to Heuston station. It takes between 35 and 40 minutes depending on the time of day.

For the convenience of the door to door service, especially when horsing luggage around, you really can't beat a taxi. Takes about 30 minutes or thereabouts
For 2 or more people it would be a cost-effective alternative to the bus.

If you're traveling solo & have 1 small luggage & a personal item, the airport bus as previously posted, is the most cost effective, although it takes a bit longer than a taxi. Plus side is you get to see parts of Dublin.
If there's 2 people then you might consider a taxi for convenience. I've generally had good experiences with taxi drivers except the last time I took a taxi from the airport I noticed he took the long way to Heuston station & charged 50 Euros instead of 40 Euros that I've paid other times. I suggest telling the taxi driver that you want a quick, least expensive way to Heuston station.