Solution 1:

In Chrome, open the console by pressing Ctrl + Shift + J. In the console, type the following and hit Enter.

document.body.contentEditable = true;
document.designMode = 'on';

This will make the page editable and the built-in spell checker of Chrome will be able to highlight anything misspelled.

Solution 2:

Use the bookmarklet from Switched.com: how-to-spell-check-any-web-site-using-firefox

You can drag-n-drop the bookmarklet link on your bookmarks bar.
You may have to drag-select the content of your page to force refresh spell check on the website's content.

Works in Chrome.

Solution 3:

I have found out a way to do it.

First you need to make the content editable:

document.body.contentEditable = true;
document.designMode = 'on';

Then as you are proofreading the page, select the text you want to spellcheck, then Ctrl+X to cut and Ctrl+V to insert it back. This should trigger the spellchecking of that block of text.