Couple of thoughts here.... First, carbon is your best bet. I also run carbon 24/7 in a filter media bag. It's extremely safe, and there's no real way to mess it up. Put it in a place where water is forced through it. I also don't have a sump, so I put mine inside my HOB filter (AquaClear 70, in my case).
A micron cartridge/polyfilter mechanical filtration has its place, but has the tendency to become a nitrate factory, unless you stay on top of frequent replacement of the media. The media will collect detritus, and all sorts of other "junk" but at this stuff sits in the media, it starts to convert to media. Mechanical media filtration is only an option if you stay on top of replacement (which means about every 2-4 days, IMO).
Ozone has the ability to clean the water, most assuredly. It WILL make your water more clear. However, ozone is more of an advanced method, and you need to make sure you know what you're getting into if you take this route. It will also be expensive, because to use ozone responsibly, you need to run an ORP controller, which will turn off the ozone generator if levels are too high. Ozone has been the culprit in many tank crashes, and you seriously need to consider what you're getting into should you choose this path.
Vodka and sugar is an even more advanced method that should only be used by those who have a real command over their tank maintenance, and are very familiar with the chemistry as well as what and how things are being affected. Said more plainly, I'd steer well clear of this method, as there are so many other options that are more foolproof and less dangerous.
UV is a nice thing to have, but you need to have a way to pump water into a UV filter and back into the tank. Without a sump, I wouldn't really consider UV an option. Maybe some day, though. I'm in the same boat as you here.
As a first step, I highly recommend activated carbon in a media bag. Fill the bag with carbon, rinse it extremely well (until the black dust stops coming out) and throw it into a HOB filter. Check it in a couple weeks to see if the bag is covered in detritus. If so, rinse well again. Replace it after a month. You may be very surprised how the clarity of your water changes.