Murphy represented his native England at under-16, under-17 and under-19 international levels, but switched allegiance to the Republic of Ireland in 2011 and played for their under-21 team.
Born in Shoreham-by-Sea, West Sussex, Murphy broke into the Arsenal reserve team as a schoolboy, while also scoring 17 goals in 21 matches for the Arsenal Academy. He was previously an academy player for Wimbledon before joining Arsenal Academy. He signed his first professional contract in July 2008. He won the 2008–09 Premier Academy League with the Arsenal under-18 team, scoring the only goal in the play-off final versus Tottenham Hotspur, and the 2008–09 FA Youth Cup.
He joined League One club Brentford on loan for three months on 24 November 2009.
In August 2011, Murphy had a trial with Scottish Premier League champions Rangers and played for their reserve team. In January 2012, he had a trial with Hibernian, another SPL club, but at the end of that month he signed for League One club Preston North End on loan until the end of the 2011–12 season.
On 22 May 2012, Murphy was released by Arsenal.
Murphy joined Dutch Eerste Divisie club Telstar on 9 June 2012, signing a one-year contract with an option for a second. He made his debut on 10 August 2012 in a 2–1 home defeat to Helmond Sport. In the 67th minute, he was substituted for Leandro Resida who scored three minutes after.
On 22 July 2013, he joined League Two club Dagenham & Redbridge on a two-year contract. He scored on his debut in the 3–1 defeat away to Fleetwood Town on 3 August 2013. He then scored his second goal for Dagenham in a 2–0 home win against York City the following week.
On 2 February 2015, Murphy signed for League One club Oldham Athletic on a two-and-a-half-year contract for an undisclosed fee believed to be in the region of £20,000, to accelerate a pre-contract agreement set to activate in the 2015 summer transfer window. Manager Lee Johnson said of the arrival, "I'm really pleased to bring Rhys to the club and he is an exciting prospect for us, I'm looking forward to working with him and he will fit into our style of play seamlessly".
He joined League Two club Crawley Town on a 93-day loan on 17 September 2015.
On 7 July 2016, Murphy signed for National League club Forest Green Rovers for a nominal fee. His first goal for Forest Green came on 20 August 2016 in a 2–1 home win over York. He followed his first goal up by scoring twice in a 4–1 away victory over Maidstone United a week later on 27 August 2016.
On 1 December 2016, Murphy joined Forest Green's divisional rivals York City on loan until 7 January 2017.
On 31 January 2017, Murphy returned to League Two club Crawley Town on loan for the remainder of 2016–17. Five days later, Murphy made his Crawley return in a 2–1 home defeat against Stevenage, replacing Enzio Boldewijn with 12 minutes remaining.
On 19 September 2017, Murphy joined National League side Torquay United on a three-month loan deal.
Murphy has played for the England under-16 and under-17 teams and was part of under-19 squad that reached the final of 2009 UEFA European Under-19 Championship.
Before playing for England, Murphy represented the Republic of Ireland at under-15 level and after an approach from Irish under-21 Head Coach Noel King, he switched his allegiance back to the country of his grandfather. He was called up to the under-21 squad in August 2011, and made his debut on 1 September 2011 against Hungary at the Showgrounds, scoring the winner in a 2–1 victory.As of match played 5 August 2017
ArsenalPremier Academy League: 2008–09
FA Youth Cup: 2008–09