Ask at the end of the appointment, while they are still pleased with the result, and ask in one sentence: 'If you are happy with it, a review really helps me.' That is the whole script. It is not awkward unless you make it awkward by apologising for asking.
Ask everybody rather than only the clients you are confident about. Selecting for the happiest people is how you end up with six reviews and a rating nobody believes, and it takes far longer to reach the point where a rating is shown at all — it stays hidden until there are at least five.
Only somebody who actually had an appointment can leave one, and only after it has happened. That is what makes the number worth something, and it is why you cannot shortcut it.
Never buy reviews, never write your own, and never offer a discount in exchange for one. Beyond being against the rules here, a page of implausible five-star reviews reads as bought to the exact clients you are trying to persuade.
Reply to the poor ones, briefly and without arguing. A calm reply saying what you offered to do is read by every future client, and it does more for you than the review did against you. Replying at length to a bad review is how a small problem becomes the first thing everybody sees.