How do you Apply Different Colour Strokes to Different Colour Letters?

So, is the color of every letter sampled from an underlying image, rather than a fixed palette? If you're willing to have a fixed palette, you could do this:

Add a Posterize adjustment layer, and set it to a value that's high enough to keep the image integrity. Select each color with the Magic Wand, set to zero tolerance with Contiguous un-checked. Or you could use Select > Color Range. Also use the Eyedropper to make the color your foreground color.

Once you have the selection, turn it into a work path via the Paths panel. Select either the Pen or Shape tool, and make the path a shape. Now, add a Stroke layer effect.

You could also do this using the selection as a mask for a Color Fill layer, but it's better to do it with vectors, so the can be re-sized nicely.

I will suggest though trying to spend some time to learn illustrator. In there you can have every character separate and editable in such a way that you can manipulate it to the Nth degree, and it's a vector image. So if you ever wanted to print one of these out at 6 feet high and keep it totally sharp, you could do it easily. Just a thought...

Do you have access to Illustrator? There, you could Select > Same > Fill Color for each color, and add the stroke en masse.

You could also keep the text on one layer, then add several clipping masks to the text layer, with the colors you want, and masked loosely around each letter.