Alessandra was born in Heywood, Greater Manchester. Along with his teammate Deane Smalley, Alessandra graduated from Oldham Athletic's youth system. He was a regular in the reserve squad, receiving a contract extension in the summer of 2008. The club awarded the youngster a three-year contract.
On 20 October 2007, Alessandra made his first-team debut in a 1–1 draw versus Huddersfield Town, with his first start coming on 2 February 2008 in a loss to Swansea City. His first goal with the senior squad came at Boundary Park in a 3–1 win over Tranmere Rovers on 8 March 2008. On 26 April 2008, Alessandra scored his second league goal in the 87th minute versus Cheltenham Town, giving Oldham a 2–1 win.
In Oldham's first match of the 2008–09 season, Alessandra scored Oldham's third goal in a 4–3 victory over Millwall. He scored his first senior hat-trick on 28 October, in a 3–0 victory over Scunthorpe United.
At the beginning of the 2010–11 season, Alessandra was converted to a wide player for upcoming season by the new manager enabling him to play up front or on either wing. At the end of the campaign his contract expired and he was informed that he would not be offered a new one.
On 31 July 2009, it was reported that Alessandra was set to join Conference Premier club Chester City on loan. The move was done with the idea of giving the player more time playing and he commented himself that "if he was not getting games with Latics, he might as well go on loan to Chester." The move was initially a trial until the club was able to sign players again. Alessandra scored in a trial match for the club against a Burnley XI and scored his second goal in two matches with an early effort against Droylsden.
After briefly returning to Oldham, Chester confirmed they had signed Alessandra on an initial one-month loan on 21 August. The following day, he made his debut in a scoreless draw versus Luton Town, playing more than 90 minutes before he was substituted for defender Kristian Platt. He made four appearances for Chester but after club was expelled from the Football Conference their record was expunged.
Alessandra signed for League Two club Morecambe on 23 June 2011. In two years with the club, he made 82 league appearances and scored nine goals playing in a number of different positions. He was offered a new contract at the end of the 2012–13 season and agreed to sign it but then decided against it and joined Plymouth Argyle.
Alessandra signed a two-year contract with Plymouth Argyle in June 2013. "I know Lewis very well, and his strengths as a striker," said Argyle manager John Sheridan, who gave him his first team debut at Oldham in 2007. "Lewis can play on the right and left of the front line as well as down the middle." Alessandra made his debut in the club's first match of the new season at Southend United, and scored two goals in a League Cup match at Birmingham City three days later. Alessandra scored his first league goal for Argyle against Northampton Town in November. In total Alessandra scored 12 goals in 49 matches in all competitions in his first season with the Pilgrims.
In November 2014, be scored his 18th goal in Argyle colours, scoring in a 3–0 win against Portsmouth. This rose Argyle up to 4th in the table, just off the automatic promotion spots. Alessandra's initial role at Plymouth was to be used as a right midfielder, where he spent the duration of the first half of the Pilgrims' 2013–14 season. Sheridan later saw the Midfielder's attacking attributes and decided to play him as a striker, alongside Reuben Reid. Alessandra ended the season with 12 goals in 49 matches, with Plymouth finishing 10th in League Two. Alessandra scored 13 goals in 51 matches the following season as his team qualified for the League 2 play offs, but were defeated by Wycombe Wanderers in the semi-final.
On 8 June 2015, Alessandra signed a two-year contract with League One club Rochdale. Having had limited chances at Rochdale, he joined League Two club York City on a one-month emergency loan on 8 March 2016. He scored in the 20th-minute of his debut in a 1–1 home draw with Barnet on 12 March 2016.
On 29 June 2016, Alessandra signed for League Two club Hartlepool United on undisclosed terms, after having his contract with Rochdale cancelled by mutual consent.
On 26 May 2017, Alessandra signed for League Two club Notts County on a two-year contract for an undisclosed fee.As of match played 16 September 2017