I visited Greece a few times while my family was stationed overseas. You definitely want to focus on hotels surrounding the Acropolis which is essentially smack in the middle of the city.
There are plenty of 4 star hotels surrounding there you can get for $100-140 a night. You can look at some of the American chains (Hilton for instance), but you're definitely going to pay more for about the same. There are smaller cheaper hotels around there as well that are still nice and clean, but you may have to forgo views and so amenities. It's mostly personal preference, but as long as you stay near the city center, you shouldn't have to worry about any cleanliness or safety concerns.
I'd definitely recommend venturing out past Athens once you've seen the Acropolis and toured the city. Olympia might be a good place to visit if seeing some of the ruins from the the Ancient Olympics interests you. If you're going to the islands, you need to go to Rodos, Rhodes. Whether you want to book a resort so you can rot on a beach or focus on the culture in the city, it has by far the most to do on both ends of the spectrum.