First of all, make sure your customer is happy
A lot of companies leave their customers alone after a sale, and that's a real mistake. “No news is good news” is not necessarily a good thing when it comes to building your reputation. Often, not following up come from fear of finding out the customers isn't perfectly happy.
So first and foremost, make sure your customer is satisfied. Under-promise and over-deliver. But sometimes customers want more than you promised. Some folks can never be made happy. And some folks get buyer's remorse no matter how good their purchase was.
A great guru friend of mine has a mantra — remain curious. It is never more applicable than here. If a customer isn't thrilled, ask questions. “What can I do?” “How can I help?”
But that aside, if you've done everything you've promised, assume your customer is happy.
Wait for it…
Timing is everything. There's nothing worse than getting a request to introduce you to a friend before I've gotten to know you. Again, assuming your customer is happy, give them a chance to use your product, or experience the results. Then ask for a referral. Gently.
“As you know, our business relies on satisfied customers like you. If you know anyone who could use our services, would you be so kind to introduce us?”