My dog has a hot spot and eye infection, can I treat it myself?

It sounds like you are describing that Archie has a flat inflamed area of skin -- this is often called a "hotspot." And if you are concerned about an eye infection I suspect you are seeing some goopy discharge form his eyes. Ideally, I would recommend you take Archie to your vet to have these problems evaluated, as they would most likely resolve fastest with prescription medications.

However, if he is comfortable--not super itchy, not squinting his eyes like they hurt, he is eating and acting normally--and you are wondering if there is anything you can do at home, you could try treating the hotspot with topical veterinary over the counter 1% hydrocortisone cream twice a day for a few days. If it is a very mild hotspot, that may be enough. As for the eyes, again if not severe (not squinting or pawing at his eyes) you can try using an over the counter eye wash made for dogs for a few days. If either condition is not improving in 3-4 days, or is getting worse, please have him seen by a vet.