How do you to check if a user has rated your app on the Google Play?

No. You cannot do this. And this is a good thing too -- otherwise you will be able to influence rating by giving people who rated it rewards and such. Additionally, developers would be able to retaliate to negative reviews if such an API were available. This might also violate certain legal agreements between the User and Google if Google starts revealing this data.

However, this is what I personally do in my apps:

1. Let the user use the app 5 times, to get a good feel of it.
2. Prompt the user on the sixth run to rate it with options for Yes, Later and Never. Later delays it by two days.

It does take an effort to get users to rate your app, especially if they like it! Unfortunately, negative ratings seem to take much less persuasion ;-)

I've tried a few things, all involving an app button titled "Rate me!" or some such, which when clicked, opens the Market App on the phone. It has been my experience that most users who click the first "Rate me" button will spend an extra 5 secs to give you a stars-rating (and remember most users don't know that your app cannot track whether they actually provided a rating or not). They may not give you a wordy review, and limit it to only a few words ("Great app"), but hey take what you can get!

Actually you can do it, you can get the username by using this or similar...

How can I get the google username on Android?

then fire a parser on https://play.google.com/store/apps/details?id=X URL...

<span class="author-name"> <a href="/store/people/details?id=11">Andy Borris</a>  </span>

<div class="tiny-star star-rating-non-editable-container" aria-label=" Rated 1 stars out of five stars "> <div class="current-rating" style="width: 20%;"></div> </div>

and then you can popup "rate me five stars or i will share your pictures with the world" :P lol just kidding

As other users mentioned before your best bet is to use a "Rate my app" button or dialog. If you use analytics you can track button clicks. If using Google Analytics you could assign event values for buttons displayed on the dialog box and see see how many users click "Rate" button.