The age old reply of 'it depends' stands true here as there is no catch all answer but only general situational guidance.
As a guideline i tend to try to initially found cities with at least 2 luxury resources within it's 3 tile ring that are not already within the ring of previously founded cities which tends to keep my cities at least 6 tiles apart but not too far apart that there are huge gaps in my borders as a continuous border prevent the AI's just plopping random cities in the middle of your empire.
If i can found a potentially huge city 6 tiles from any nearby city but that would mean i only get 1 resource instead of founding that city 5 tiles away and obtaining 2 resources i will found it 5 tiles away or even 4 although 4 tiles i generally consider too close really.
Having said that in my current game i founded a satellite city 4 tiles from my capital to obtain a marble and 6 iron resource node which i would be able to obtain with my capital and i couldn't found it further away due to a mountain range so while i wouldn't generally plan for a city 4 tiles away i will happily build one 4 tiles away if i deem it to be worth it.
It also depends on your general strategy. If your going for a liberty land grab/city spam style of empire then you will likely have lots of smaller cities due to early to mid game unhappiness issues so keeping them closer means you can start building roads earlier and can quickly grab more shared resource tiles that multiple cities can use depending on need to compensate for reduce border growth.
If your going for a small and tall tradition style empire then you want to try to space them further apart to give each city the maximum number of workable tiles for the high pop you should eventually have and because you should have rapid border expansion so should fill in any border gaps quite quickly.
Again though you need to be flexible and use those as general guides. In BNW the old ICS style of build a block of cities all 4 tiles apart isn't that effective if some of those cities are purely space fillers.i.e. don't obtain any new resources as they simply become a drain on your empire until at least the mid to late game so in the early game where your usually playing catch up anyway you are only making the hill steeper. If the city at least obtains a new resource you can then at least use that resource to remove some of the penalties from building new cities such as selling it so you can purchase buildings you need etc.
If your going tall then discover a big clump of a strategic resource just outside your border range like a clump of iron nodes then it can be feasible to plant a satellite city at 4 tile range if needs be.e.g. your blocked in by terrain or other cities.
So in summary look at the potential city spot and what you will obtain by planting it...
- strategic location (it could give you access to the sea so you can build ships, create a choke point or just create a heavily defensible barrier with an aggressive neighbour
- potential gold income (early game this also means being able to send a trade routes either to a nearby neighbour or simply gaining access to sea routes)
- access to resources
- will it be able to grow
- does it need to grow
- will it interfere with any other cities
- does it have a good chance of providing you with future resources (oil and uranium are often most common in traditionally poor city locations such as tundra or desert)
- etc
then weigh that against any negatives you might get...
- will it annoy your neighbours
- do care if it annoys your neighbours, may actually want it to annoy your neighbours.e.g. provoke them into DoW you or to ensure they don't obtain a resource.
- each city increases your social policy cost and tech cost as well as being a general drain on empire happiness
- if it also won't produce a lot of gold then any building you may need to build to make it worthwhile may also be a drain on your gold income
- will it be vulnerable to attack
- etc