Alexander Claffy was born June 9, 1992 in Philadelphia and grew up inside the city's working music scene rather than at a distance from it. His father led a band, and by his own account Claffy's earliest lessons came on bandstands in Philadelphia and New York rather than in practice rooms. He started on electric bass and moved to the upright, studying with the Philadelphia bassist Mike Boone and later with Ron Carter, Dwayne Burno, Gregg August and Orin O'Brien. He moved to New York in 2011 to attend the New School for Jazz and Contemporary Music, where Jimmy Cobb was among his teachers, and he has since returned there as an adjunct instructor.

His recorded career begins in his early twenties and moves quickly. The first documented session is January 29, 2013 at Systems Two in Brooklyn, for pianist Orrin Evans's Criss Cross album "...It Was Beauty", where Claffy plays a single track rather than the whole record. A year later, on January 8, 2014, he took the full bass chair on tenor saxophonist J.D. Allen's Bloom (Savant), with Evans on piano and Jonathan Barber on drums. Through the middle of the decade he became a fixture of the younger New York studio scene: two Posi-Tone albums for trombonist David Gibson, Boom! (2015) and Inner Agent (recorded January 17, 2016), guitarist Ben Eunson's Autumn and ACE, and trumpeter Benny Benack III's One Of A Kind (2017), a band that also held Joel Frahm on saxophone and Yotam Silberstein on guitar.

Claffy's first record under his own name, Claffy, came out on Ropeadope in February 2016 and is not a conventional bass-led date at all. It runs as a nine-part suite of an introduction, six numbered "Chapters" and an epilogue, with two credited narrators, four singers including Camila Meza and Dida Pelled, and Claffy himself on bass, keyboards and percussion. Standards: What Are You Doing the Rest of Your Life? followed on SMK Jazz in 2018 and went the opposite way: an unhurried, largely trio-plus-guests reading of nine standards with Kurt Rosenwinkel on guitar, David Kikoski on piano, Joel Frahm on tenor and Veronica Swift singing "Deep Song." Ron Carter wrote the liner notes for his former student. Claffy II (Ring Road, November 2019) split the difference, five tracks with five different drummers, Eden Ladin and Addison Frei sharing the piano and Josh Evans on trumpet.

The Smalls and Mezzrow circle in Greenwich Village runs through the whole of his New York work. On November 19, 2017 he played bass in the Charles Owens Quartet at Smalls alongside Joel Frahm and Ari Hoenig; the resulting Live At Smalls (SmallsLIVE SL-0057, 2018) was produced by the club's managing partner, Spike Wilner, and includes three collectively credited "Interludes" and a "Goodnight Blues" written by the four players together. In October 2020, at Big Orange Sheep Studios in Brooklyn, Claffy cut three duo tracks with tenor saxophonist Nicole Glover: "One For Amos," "Spring Can Really Hang You Up the Most" and "Good Bait." Those three became side B of The Complete Orange Hours (Diggers Factory, 2021), a 2xLP that gathers four unrelated duos, one per side, and issues each duo's three tracks separately as its own LP345 Records single.

From 2022 his leader catalogue settled onto Cory Weeds's Cellar Music. Good Spirits (March 2022) is a straight-ahead quintet date with Nicole Glover on tenor, Benny Benack III on trumpet, Julius Rodriguez on piano and Joe Farnsworth on drums, opening on McCoy Tyner's "Inner Glimpse" and including an original called "For The Great Gerald Cannon." Music from Big Orange followed in October 2022, named for the Brooklyn studio and carrying two pieces of Philadelphia in its titles: "Cheltenham," for the township north of the city, and "1421 Sansom," the street address of Chris' Jazz Cafe. Memento, recorded at Big Orange Sheep in February 2023 and released in June 2024, put Kurt Rosenwinkel and Jimmy Macbride behind a rotating front line of Simon Moullier's vibraphone, David Kikoski and Timothy Brey on piano, and Stacy Dillard, Matt Chalk and Eric Alexander on saxophones. The same year he took the bass chair in the Heavy Hitters, the Mike LeDonne co-operative with Eric Alexander, Vincent Herring, Jeremy Pelt and Kenny Washington, recorded live at Frankie's Jazz Club in Vancouver over three nights in December 2023 for That's What's Up.

He has also kept a parallel life as a sideman with singers and with leaders well outside his own generation, from Randy Brecker and Ada Rovatti's Sacred Bond (2019) to Kristiana Roemer's House of Mirrors, Chad Lefkowitz-Brown's Quartet Sessions, Seamus Blake's New York Factor Vol 1, two tracks of Veronica Swift's 2023 Mack Avenue album, and Kerry Politzer's Alternate Route in 2025. In March 2025 he brought a quintet of Jaleel Shaw, Seamus Blake, Kevin Hays and Bill Stewart home to Chris' Jazz Cafe for two nights, and released the results as Alive In Philadelphia Vol. 1 on Cellar Music in March 2026, his first live album as a leader and the record that returned him to the city he still works to support.

Claffy plays with a big, dry, centred sound and an unusually strict sense of time; the marlbank review of Alive In Philadelphia Vol. 1 praised him for measuring the beat meticulously through a whole live set, and he is most often hired precisely for that steadiness, which is why bands as different as Mike LeDonne's Heavy Hitters and Kurt Rosenwinkel's quintet call him. What distinguishes him from other reliable straight-ahead bassists of his generation is how little interest he has in staying inside that lane on his own records: Claffy is a narrated suite with four singers, Claffy II uses five drummers across five tracks, and Mostly Mono Mixtape is not a jazz record at all. He grew up with Philadelphia's bass tradition behind him, the line that runs through Percy Heath, Jimmy Garrison, Christian McBride and his own teacher Mike Boone, and his stated listening runs from John Coltrane and Miles Davis to Radiohead and Cream, a range his leader dates make audible.