They're sitting there in the cantina, both drawing their weapons without the other realizing it, or perhaps knowing each other so well they expect exactly this from each other. Does it matter, then, which of them shoots first? Suffice to say, it's a lethal proposition, and they both know it, and it was a long time coming, story of their lives, not just because of what they did for a living and who they worked for, but because, well, they just don't like each other.
And at the end of it, Han Solo was dead and Greedo sat there, stunned that it had finally happened. He'd finally bested his old rival. As far as he could tell, Solo had misfired somehow, perhaps since Greedo had gotten his shot off first. He thought momentarily that it was a tainted victory, but he was satisfied all the same. He stood to get up, and realized his knees were giving out on him, whether a matter of age or nerves or the shock of the moment.
And then he realized he didn't know where he was going. The funds would be transferred to his account as soon as the body was delivered, but suddenly that didn't seem like such an urgent matter. He went to the bar to pay his tab, and he actually flipped the bartender enough to cover both their drinks. "Sorry about the mess," he said as he turned, trying to play it cool, the way he imagined Solo might. Truth was, he'd always admired the man, and in fact envied him.
Where was that ship of Solo's? The thought came into Greedo's head completely unbidden. Why did he even care? Maybe he was looking to confiscate it? Claim it as his own? He found it in a hangar bay. There were Stormtroopers crawling all about. Now he was intrigued. His old friend had been in serious trouble, it seemed. He snuck around and entered through the top hatch, climbing upward and shaking his head as he went. What are you thinking, Greedo?
Once aboard, he came across the Wookiee. They knew each other well enough, had even worked together. They both knew that this day had always threatened to happen, and Greedo once again felt a curious sensation in the pit of his stomach, something unsaid telling him that his life was in the process of changing forever. He noted vaguely that there were passengers aboard.
"Punch it," he said as he settled into the cockpit. It was crowded; he didn't acknowledge anyone else. The only thought he had was of escape, now. The Wookiee complied without comment. It was the first time Greedo could remember that the Wookiee chose silence, probably out of respect for his old partner. At least he knew where they were going. Greedo was content to go for the ride.
When they reached the debris field, he became curious for the first time. Everyone knew that these were the coordinates for Alderaan. What had Solo been a part of? The Rebellion, it occurred to him for the first time. It reshaped the course of everything Greedo thought he knew. He forgot about the money. The Imperial space station pulled them in with a tractor beam, one of the passengers said, "I have a bad feeling about this," and a droid let loose some nervous chatter. It was reaching the point where there was no turning back.
But Greedo had already made his decision. He was going to resume Solo's life for him.