There's no such thing as objectively good work. It's only good according to how well it fills a need. That theme permeates your drive to deliver quality work, and it's your client who determines that your work is quality. Here are some ways to not only do the work well, but to help them see that the work was done well. I take steps that follow the old saying tell them what you're going to do, then do it, then tell them what you just did. Or, in other words, confirm the task, deliver the goods, and review what happened.
Let's start with confirmation. It's not just a one time thing. You'll have to keep asserting your understanding throughout the project. Whenever you get new information from the client, confirm not only what it is, but how it will be incorporated into the project, and when they will see the results.
Then comes the second step which is the biggest: deliver the work. But there's more to it than that. The delivery has to fit their expectations according to the six traditional questions:
First is who. Are you delivering the work to the right person? If you're not sure, confirm it. Next is what. Are you delivering what they asked for? This should be spelled out in your agreement. Then is why. Aside form what's in the agreement, does the product satisfy the reason you were hired in the first place? Is anything else needed to of your work truly useful? The questions of where, how, and when should also be answered in the agreement. Do they want a printed copy, a digital version, both, or something else? Does a physical object need to be delivered? The more your delivery matches their expectations, the more likely they will be satisfied and that they will look forward to hiring you again.
So that takes us through the confirm and deliver phases. Now for the review. Even if, all the work is done, you need to make sure that the client is satisfied. If they are, then give them the satisfaction of closure by letting them declare it done. If they're not satisfied, sincerely listen to their concerns and review the project's details. But let's assume that everything went well. While there's a good feeling between you, take this moment to ask them to memorialize that with an endorsement. That reminds them of how satisfied they were while also giving you a powerful tool for marketing your services to others.