If you have the phone and he is using the standard Chrome browser, you can open the browser, then open the menu at the top right (three vertical dots). Click on history and you can see the browsing history of all signed in devices on that account. If he is using a different browser, it still probably has a history setting. Keep in mind that any browsing done in incognito mode will not be saved.
I don't know of a way that you can see what someone wrote unless there is a key logger already installed on the device.
Word of warning, though. I would personally be pretty upset if my wife was going around and monitoring everything I did. I respect her privacy and would never sneak behind her back and go through her things, and expect the same measure of respect from her. You may have your reasons, but be aware that you might be opening a can of worms here. Good luck.