How to get a pixel's x,y coordinate color from an image?

Building on Jeff's answer, your first step would be to create a canvas representation of your PNG. The following creates an off-screen canvas that is the same width and height as your image and has the image drawn on it.

var img = document.getElementById('my-image');
var canvas = document.createElement('canvas');
canvas.width = img.width;
canvas.height = img.height;
canvas.getContext('2d').drawImage(img, 0, 0, img.width, img.height);

After that, when a user clicks, use event.offsetX and event.offsetY to get the position. This can then be used to acquire the pixel:

var pixelData = canvas.getContext('2d').getImageData(event.offsetX, event.offsetY, 1, 1).data;

Because you are only grabbing one pixel, pixelData is a four entry array containing the pixel's R, G, B, and A values. For alpha, anything less than 255 represents some level of transparency with 0 being fully transparent.

Here is a jsFiddle example: http://jsfiddle.net/thirtydot/9SEMf/869/ I used jQuery for convenience in all of this, but it is by no means required.

Note: getImageData falls under the browser's same-origin policy to prevent data leaks, meaning this technique will fail if you dirty the canvas with an image from another domain or (I believe, but some browsers may have solved this) SVG from any domain. This protects against cases where a site serves up a custom image asset for a logged in user and an attacker wants to read the image to get information. You can solve the problem by either serving the image from the same server or implementing Cross-origin resource sharing.

Canvas would be a great way to do this, as @pst said above. Check out this answer for a good example:

getPixel from HTML Canvas?

Some code that would serve you specifically as well:

var imgd = context.getImageData(x, y, width, height);
var pix = imgd.data;

for (var i = 0, n = pix.length; i < n; i += 4) {
  console.log pix[i+3]
}

This will go row by row, so you'd need to convert that into an x,y and either convert the for loop to a direct check or run a conditional inside.

Reading your question again, it looks like you want to be able to get the point that the person clicks on. This can be done pretty easily with jquery's click event. Just run the above code inside a click handler as such:

$('el').click(function(e){
   console.log(e.clientX, e.clientY)
}

Those should grab your x and y values.

The two previous answers demonstrate how to use Canvas and ImageData. I would like to propose an answer with runnable example and using an image processing framework, so you don't need to handle the pixel data manually.

MarvinJ provides the method image.getAlphaComponent(x,y) which simply returns the transparency value for the pixel in x,y coordinate. If this value is 0, pixel is totally transparent, values between 1 and 254 are transparency levels, finally 255 is opaque.

For demonstrating I've used the image below (300x300) with transparent background and two pixels at coordinates (0,0) and (150,150).

Console output:

(0,0): TRANSPARENT
(150,150): NOT_TRANSPARENT

image = new MarvinImage();
image.load("https://i.imgur.com/eLZVbQG.png", imageLoaded);

function imageLoaded(){
  console.log("(0,0): "+(image.getAlphaComponent(0,0) > 0 ? "NOT_TRANSPARENT" : "TRANSPARENT"));
  console.log("(150,150): "+(image.getAlphaComponent(150,150) > 0 ? "NOT_TRANSPARENT" : "TRANSPARENT"));
}

<script src="https://www.marvinj.org/releases/marvinj-0.7.js"></script>