This time period is addressed in the book Rebel Dawn.
Han Solo is on Tatooine at the beginning of A New Hope because he just recently tried to meet with Jabba to discuss the fact that he had been boarded by Imperials and had to jettison his cargo of glitterstim. Jabba, however, was unresponsive and unwilling to meet with Han.
Han was still attempting to get a meeting with Jabba to avoid conflict, or to acquire enough money to pay off Jabba. Jabba, however, decided to send bounty hunters to collect the money or Han's corpse.
Until Greedo showed up, Han was doing exactly the opposite of trying to avoid Jabba - he was specifically trying to get a chance to actually talk to him. When Greedo shows up, Han initially implores Greedo to pass along Han's message. Instead, Greedo pulls a blaster. Unfortunately for Greedo, he never gets a shot off before Han shoots him first.
With a bounty on his head, Han realizes that Jabba isn't willing to hear him out and decides to get out while he can, and accepts a fare from some old guy, a boy, two droids and no questions.