Born in Liverpool, Melia was a schoolboy international of exceptional promise and clearly destined from an early age to have a long and distinguished football career which he certainly did by playing in over 500 Football League games for five different clubs. Melia was born in the Scotland Road district of Liverpool and had five sisters and five brothers of which he was the only one to pursue a football career. He was chosen captain of the Liverpool Schools side and made the England Schoolboy team starring alongside Bobby Charlton and featuring later for the England youth team. Melia signed professional forms with his hometown club on his seventeenth birthday in November 1954. Liverpool had just been relegated to the Second Division and were desperate to return to the First, especially as city rivals Everton had passed them on the way down returning to the top league. Melia had to wait until shortly before Christmas 1955 to make his debut where he delivered a man of the match performance setting up one goal and scoring Liverpool's second in a 5-2 victory over Nottingham Forest. He played in four games out of five before he had to return to his National Service. First-team opportunities were curtailed but he settled for appearances with the British Army side as well as Liverpool's reserves. In the following season, only 19 years old, he broke into the first team taking over from John Evans. Inside-left Melia flourished and was Liverpool's top-scorer in the 1958/59 season, scoring 21 goals in 40 games. He worked hard for the team and possessed a good football brain.

The strain of just missing out on promotion three seasons running finally took its toll on manager and ex-player Phil Taylor and it quickly became apparent that Bill Shankly saw in Melia all the qualities that were needed to help the club back to its position amongst the country's elite. Following two disappointing defeats at the start of his tenure Shankly moved Melia from his natural position at inside-forward to the right wing. Shankly explained: "This was a case of expediency, as we really had not wings of sufficient experience among the juniors to play in the senior side. The plan was for him to be a deep-lying inside-right/outside right with Roger Hunt playing the role of a poacher at centre-forward alongside Dave Hickson." Melia struggled on the wing and was replaced by Billy Liddell, who was fast approaching the end of his career. On his return to the side two months later James Harrower made way as Melia was put in his old position. Youngster Ian Callaghan was given a go on the right wing in the last month of the season. Melia played in all 42 league games when the team took the second division title in 1961/62.

Melia had a good debut season in the First Division, missing only three games, but in the middle of December in the following season he got injured. It was his first real injury and dented his confidence, in fact he only played three more games for Liverpool before he lost his place to Alf Arrowsmith. He was almost immediately transferred to Wolverhampton Wanderers, although he had played in enough league matches before the transfer to qualify for a championship medal. Before the end of the year though, he had moved further south to join Southampton and he enjoyed four good years there, including helping them into the First Division in 1966, before he moved on to Aldershot as a player-coach, later taking over as manager. In January 1971 he had the enormous thrill of leading his club out for an FA Cup third round tie at Anfield and he received a wonderful reception from the Liverpool crowd who remembered how tirelessly he had worked when he was one of their own.

After his playing days were over, Melia coached in the United Arab Emirates, Kuwait and USA before being appointed Brighton & Hove Albion's chief scout, becoming their manager in the 1982/83 season. After Brighton's relegation to the Second Division in 1983 he moved to Portugal where he managed Belenenses for three years. He later moved to America where he served as Technical Director of Liverpool's Academy in Texas. In his post-Liverpool career he is probably most remembered for being behind Brighton's heroic run to the FA Cup final in 1983, which included a shock fifth round victory over Liverpool at Anfield when they were bottom of the First Division and their hosts were top. To make matters worse, the winning goal was scored by a Liverpool old-boy, Jimmy Case!