Byrne, the first of six children, was born in Dublin, Ireland, the son of a cooper, Dan, and a hospital nurse, Eileen (née Gannon). His siblings are Donal, Thomas, Breda, Margaret, and Marian, who had passed away at a young age.

Byrne was raised Catholic and educated by the Irish Christian Brothers. Despite spending five years of his childhood in a seminary training to be a priest, he said in an interview, "I spent five years in the seminary and I suppose it was assumed that you had a vocation. I have realised subsequently that I didn't have one at all. I don't believe in God. But I did believe at the time in this notion that you were being called.".

He attended University College Dublin, where he studied archaeology and linguistics, becoming proficient in Irish. He played football in Dublin with the famous Stella Maris Football Club in Drumcondra and has fond memories of his time spent there.

Byrne worked in archaeology when he left UCD but maintained his love of his language, writing the first drama in Irish, Draíocht, on Ireland's national Irish television station, TG4, when it began broadcasting in 1996.

He discovered his passion for acting later in his life. Before becoming an actor, Byrne worked at everything from an archaeologist to a cook, to a bullfighter to a Spanish schoolteacher.

When he finally found acting aged 29, he began his career on stage with the Focus Theatre and the Abbey Theatre in Dublin, later joining the Royal Court Theatre and the Royal National Theatre in London.

The actor came to prominence on the final season of the Irish television show The Riordans, subsequently starring in his own spin-off series, Bracken. He made his film début in 1981 as Lord Uther in John Boorman's classic King Arthur epic, Excalibur.