If the cat approached you, I would say the cat doesn't have an ingrained fear of humans. They could be an outdoors cat, or lost or dumped... it's hard to tell. But if you're familiar with the area, and haven't seen it before, then the cat may be new.
If you can get close enough to the cat (probably without your dog
) snap a picture and check shelters, local vets or animal control to see if the owner has reported them missing. I know my local shelter posts lost cats & dogs for the area all the time on Facebook. You can also run ads in the paper, the paper I work at does this for free - but some papers may charge. Ask your neighbors too, cats don't typically stray too far from their home if they just got out. They might also know if the cat's been around for awhile, or if someone else takes care of it, etc.
If you get on good enough terms with the kitty, try and feel the back of the neck for a microchip. It'll be like a grain of rice. If the owner had their cat microchipped, have a vet or an animal shelter scan the cat to see if the owner registered their chip. That'll give you contact information.
Actually, if a cat looks really rough, matted etc... that could still be a recently dumped/lost cat. Indoor cats don't know how to handle the elements like a cat that's lived it's life outside - they're scared and there's all this nasty weather and dirt they don't know how to deal with.
So they can look pretty rough, although this is easier to tell on long haired cats. This is what a shelter volunteer told me once. I know a cat I rescued that was dumped was covered in mats, and dirt. After I took him in - he was gorgeous from then on. Course he had a friendly human groomer after that.
Feeding a cat is a good way to earn trust - just know that once they know where food is, they will continue to show up for it. And other critters can turn up too for a free meal. I've had coons and opossums show up on my deck.
That depends on where you live though. heh. And if the cat is intact... well that's a whole 'nother bag of problems. Kittens, yowling, cat fights, spraying... etc... Our shelter always says being a feeder is okay, but not to be a breeder.
If you are convinced its a lost/abandoned cat, and cant find the owner and don't want to keep it for yourself or cant find anyone, you can try and contact a shelter for help. Do try to find a no-kill shelter, that'll give the cat the best odds (most people want to adopt kittens/puppies and overlook older pets). I want to say 70% of cats brought in to shelters are euthanized. The shelter may not be able to take them, though. Many just don't have space.
I hope it all works out for you and the cat. These are just suggestions if you keep seeing the cat around and want to help. I didn't mean to get so wordy. Sorry!