Staying airborne for extended periods is currently limited to ultralight solar powered craft (clearly not an option for carrying a city) and modern zeppelins.
The only way to come even close would be to have a fleet of these zeppelins carry the city, which would have to be ridiculously light in construction. Loading it up with all the stuff humans consider necessary would most likely prevent it from taking off, so it would be an extremely unattractive place to live. Also, unless it's anchored over water, the liability insurance would be ruinous.
Let's try some numbers. The linked article mentions prices for two models and their carrying capacity, per YEAR:
A model of the zeppelin which carries 66 tonnes will cost more than
\$25 million per year, while a 250-tonne version will be \$55 million.
But what would a 5000 person city weigh? Let's start with the most mobile homes we have, RVs. These weigh 6-10 tons a piece and "house" 2 people. They include the engine, frame and wheels of course, so let's halve that and be optimistic: 3 tons per 2 people, or 7500 tons for 5000 people.
But that's only houses, there is no infrastructure, offices, schools, control systems etc. Let's add 2500 for infra and then 2500 more for reserve lifting capacity, for 12,500 tons total.
If we use the 250 ton zeppelin we need 50 of them or a cool $ 2.7 billion per year just for the zeppelins...
Please note that you would not actually solve the real problems caused by overpopulation, making them worse instead. The earth is overpopulated because of the resources humans need/use, and those are exactly the things you exclude from the city: production of food, clean water and energy as well as dealing with all the pollution caused by those activities. As far as housing people goes, it would be easy and much cheaper to build more skyscrapers on the ground, there is still plenty of space for that.