Krillin at the point of the warning was already stronger than Ginyu Saga Goku, so, taking into account Goku's last zenkai, I don't think he'd need more than one senzu bean to reach the millions. Over the following years, he eventually catches up with the top fighters, at least in base form, though I don't think he would become so obsessed with the transformation as to achieve it during the 3 year period the way Vegeta did. Alas, he remains a non factor during the first part of the saga, during which the events go as we know.
Most likely, he gets angry enough to unlock SSJ when Cell absorbs Eighteen. He attacks Cell in his new form and gets stomped all the same. When the Cell Games are announced, he goes into the spirit room, the first time alone to transcend the SSJ form the way Goku/Gohan did, the second time with Trunks as a sparring partner. I'm not sure whether he can fully master the SSJ as Goku, but, considering the OP implies he is the most skilled fighter, let's say he does. Therefore, by the Cell Games, he's at least close to Goku's level. Trunks also is considerably stronger than in canon.
When Cell releases the Cell Jrs, Krillin and Trunks manage to kill them before they do serious damage, so Gohan does not get mad. Seeing how powerful Trunks and Krillin are, Cell decides to go full power and attack the Z Fighters himself. With Cell being busy with that, Sixteen's head does not die, though someone has to die for Gohan to go SSJ2. For simplicity sake, let's say it's his closest friend, Krillin.
Gohan goes SSJ2 and the fight goes as in canon. With Krillin not being there, Vegeta kills Eighteen while she's unconscious. When Cell returns, he gets killed as in canon.
Shenron can revive neither Goku nor Krillin, but they both are cool; Goku wants to train in the Otherworld and Krillin starts a relationship with Eighteen, who also is dead. As a result, he does not train as hard as Goku, though he still unlocks SSJ2 during the 7 years.
Seven years later, Goku and Krillin both return, with Eighteen remaining in the Otherworld. Krillin follows the rest of the Z Fighters to Babidi's lair. Back in the tournament, Mighty Mask (Goten/Trunks) defeats everyone (including Mr Satan) before his identity is exposed and wins the tournament.
At Babidi's spaceship, Krillin dodges Dabura's saliva and gets in the spaceship too. He takes Gohan's place against Dabura and wins. Goku and Krillin return to the Otherworld.
7 years later, Beerus does not get angry, though Bulma still befriends Whiss. Eventually, Vegeta becomes his student and the next God candidate. Freeza is defeated by Vegeta, with the help of Whisses time reversal. U6 wins the U6/U7 tournament. No Zamasu Saga. Perhaps, when Zeno appears at the tournament, he befriends Vegeta instead of Goku and Vegeta eventually somehow convinces him not to erase any universes.
