This would typically be done with AJAX. You can write an AJAX request manually in JavaScript, but it's usually better to use an existing library (e.g. jQuery) to handle this for you.
Below is an example that uses jQuery's post() to post individual values to the imaginary page named somepage.php:
$.post("somepage.php", {
name: 'John',
age: 28,
sky: 'blue',
pants: 'on'
}).done(function (data) {
alert("Data submitted: " + myArray);
});
If for some reason you still want to use your json string and post it, you could pass it as a single value named 'data':
var myArray = ["john", "28", "theworld", "blue"]; //javascript array
myArray = JSON.stringify(myArray); //to JSON
$.post("somepage.php", {
data: myArray
}).done(function (data) {
alert("Data submitted: " + myArray);
});