What's the best way to clean under a dogs eyes?

This is very common, especially with little white dogs like Maltese. It is often a sign that your dog is allergic to something in his diet. One common allergen is wheat - a common ingredient in dog food. Others are soy and corn. Try giving your dog a high-end grain-free food for a couple of weeks; if the tearing subsides, even somewhat, there's your answer. Dogs who have food allergies also will lick their paws after eating, and this can make their paws look stained, just like their faces.

One of the best ways to clean under a dog's eyes, is to buy some animal cleaning wipes and gently hold your dog while you carefully clean underneath the eye with the wipe. *maybe even twist it into a cone shape to make it safer and easier to clean. *if your dog is upset and doesn't like this make the process quick, and reward your puppy with a treat afterwards! HOPE THIS HELPS!!!

Try a bath or shower first....

I once had a rat who looked like he was crying what almost looked like blood every time I took him outside. (A big fat drop would appear in each eye almost instantly.) Turns out it was just allergies. I'd imagine there's a good chance it's the same thing with your dog, especially if it seems to be a seasonally-affected thing. I bet there's a relatively inexpensive allergy medication you could get from a vet. It'd at least be worth checking it out.

He has been to the vet for his annual checkup and nothing's wrong with him.

Well I would take your dog to a vet............