Axelrod was born to Russian-Jewish immigrant parents in New Jersey. His father was a mathematics and violin teacher, and his mother was a civilian employee of the U.S. Navy.

While in Korea he wrote his book The Handbook of Tropical Aquarium Fishes. After returning from Korea, Axelrod earned a Ph.D. at New York University and started the magazine Tropical Fish Hobbyist. He wrote many other books on tropical fish and founded a publishing firm, TFH Publications (named for the magazine) that became the largest publisher of pet books in the world. TFH Publications was headquartered first in Jersey City, New Jersey, then in Neptune, New Jersey.

In 1989 he donated his collection of fossil fish to the University of Guelph, which the university says is one of the largest donations by an individual to a Canadian university. The Axelrod Institute of Ichthyology at that university is named for him. Leonard P. Schultz discovered the cardinal tetra, a popular aquarium fish, in Brazil in 1956, and gave it its scientific name, Paracheirodon axelrodi, which honors Axelrod.

In 1997 Axelrod sold TFH Publications to Central Garden & Pet Company of California for $70 million. The contract included potential payouts to Axelrod if TFH reached earnings targets after the sale. He sued under that provision, accusing Central Garden of suppressing earnings to avoid paying the extra money. The following year, however, the purchaser filed a countersuit against him, claiming that he had grossly and illegally inflated the value of the company before the purchase. On September 1, 2005, Axelrod was ordered to pay Central Garden & Pet Company $16.4 million (net, after deducting $3.7 million the company was ordered to pay Axelrod due to earnings targets).

Axelrod, a violinist himself, assembled a large collection of old and rare stringed instruments, including the Hellier Stradivarius. In 1975 he bought his first Stradivarius violin. In 1998 he donated four Stradivari instruments — two violins, a viola and a cello — to the Smithsonian Institution. Known as the Axelrod quartet, their value was estimated at $50 million. In February 2003 he sold about 30 other instruments to the New Jersey Symphony Orchestra (NJSO) for $18 million. This collection was estimated to be worth $49 million. (Axelrod was a long-time supporter of the NJSO.)

Questions surfaced about the value of the instruments he had donated to the Smithsonian and the NJSO. Although the instruments were all old and valuable, several were determined not to be the instruments he represented them as, and not to be as valuable as he claimed. He was said to have invented histories for the instruments to explain away doubts or to increase their worth. On April 13, 2004, Axelrod was indicted in federal court in New Jersey, accused of funnelling millions of dollars into Swiss bank accounts over 20 years without paying taxes. The following April 21, he failed to appear for his arraignment, having fled to Cuba. He was arrested in Berlin on June 15, 2004 as he got off a plane from Switzerland, and then extradited to the United States. On March 21, 2005 he was sentenced in U.S. court to 18 months in prison for tax fraud.