I play for quick science victories for which the classical Great General is perhaps the most essential unit which makes swordsman/horses super powerful and relevant.
Whether or not I will go for the GG depends primarily on: *the terrain around the capital and the flow in the first 10-20 moves in which the decision is made (including the barbarian activity). *the civ and its unique units/abilities and what other civs are doing - and finally the difficulty.
If i decide to go for the GG - I will get the GG

- as I do not want to allow/risk going for the GG if I am not sure I can pull it off. This means there are situations in which I will discard the idea. So what follows is a quick General Strategy when going for the GG, and below is a list of conditions under which I will
not go for the GG.
Classical Great General Strategy (with any civ):
* Most of my normal opening will start with a builder (assuming there are 2-3 tiles that will benefit from improvements) or a scout (if immediate improvements are not viable/productive (e.g. too much jungle etc) and/or the terrain is particularly hard (too many hills/forest around the city). After that I always build a settler (to guarantee quick political philosophy)
* My normal civic progression is to put out the second city and then boost Craftmanship. Only then I go for producing the army.
IF barbarian activity is such that in the first 20ish turns I get 2-3 of them (and have a reason to believe I will get another on in the next few turns) - then TECH-wise, after perhaps maximum of Mining/Animal Husbandry and 1/2 Archery I will switch into Bronze Working (or BW first then archery - depending on the boosts). CIVIC wise i will go into Military Tradition after Craftsmanship (but not finish it). In any case - I will make sure not to finish researching more than 3 civics - and will be switching across civics waiting for the boosts.
With only the above TECHS/CIVICS researched - an early encampment will cost less than 80 production, a regular capital at this point will have probably 4 pop and good production - so making the encampment will not take more than a few turns. If I have a builder available -- (e.g. goody hut/ or I may purchase an extra one if gold is aplenty (e.g. commercial city state) // or rarely - skip the Craftmanship boost (if the terrain is not good and I find one or two Cultural City states early enough) thus saving builder charges // or more likely - because I stole one (from the first neighbour or a city state)) -- then I will use chop overflow from Agoge (50% production on military) to speed up the encampment.
Then beeline Political philosophy (which, with all the boosts) should come anywhere in the 40 (preferred) to 50 turns range. Go for Oligarchy and slot in the +2GG points card immediately.
With this strategy - the encampment will be up around the 30 (plus maybe a few turns mark). To get the GG at around turn 60 or so I will also probably run one Encampment project as well.
If you're playing with 'special' civs - then adapt.Gorgo for example has a wildcard - so the best way is to simply skip an encampment and neglect an early barb camp so that you get overrun by a barb invasion. Then every warrior you kill will get you 10 culture. So just farm barbs for invaluable early extra culture while keeping the GG card in position.
* If you were able to snatch the Goddess of the Harvest Pantheon - then you will probably be likely to finish up the GG with faith.
* If you do snatch this pantheon - and you're playing on lower difficulties (prince to emperor), and you hit early PP,w and no other civ is earning GG points when you do - then you could simply skip the encampment, late slot the +2 GG points card and finish the GG with faith.
When not to go for an encampment:
The higher the difficulty the lower the chance of success. Keep in mind that I'm writing this from the perspective of quick science victories (140-150 turns) and not domination/culture ones.
*If playing immortal/deity and you see that another civ is already earning GG points (e.g. Gorgo may slot the GG card / or very often Hojo will go for an early encampment) and you can't do an early rush on their cities - just give up on the GG - because you will most likely loose the race as the AI will just purchase the GG with gold if you overtake with the number of points.
One way around this - is to mess with their gold productions - e.g. if conditions allow - declare a war on this civ (and send a couple of units) as this may make them spend the gold on purchasing units and thus not have enough for the GG (worked for me once, didn't twice).
Alternatively - if you're in a position - to chop 2 successive Encampment projects and pay the rest in gold - great (this is possible, if you have survived an early attack from another civ - who will offer you lucrative peace deals).
*If in the first 20ish turns you are not 'lucky' with barbs -- and you are not able to clear a barb camp and get at least two of them and have a reason to believe a third one will come soon - so that the boost for Bronze Working is coming too late -- then waiting for the encampment (to boost your State Workforce) will delay the Political Philosophy too much and will backfire.
I will just note that - any other slow strategy - where you go for an encampment later, or encampment as your second district etc - then you will most likely miss the classical GG, unless if you're playing on Prince/King or none (none?) of the other civs is building an encampment (quite rare).
Hope this helps.