Is a cultural win viable in Civilization?
Cultural victory is possible, but in my experience is the hardest of the victories.  Technology victory is just a time limit on the game and not in and of itself interesting.  I have found military and economic victories to be much easier.  However, I have seen cultural victories happen.  Here are factors that make it easier:


Have a helpful civilization.  Rome looks the best on paper, though Egypt can actually be better both due to the starting wonder, which can help kick-start culture production, but also from it's ability to build free buildings which lets you build at least one city that is entirely devoted to culture production (other civilizations can't do this anywhere near as efficiently because such a city is surrounded by temples, universities, trading posts, great people, and a wonder, and as a result is deprived of hammers with which to build these buildings).  America is also okay in their ability to get wonders easier (from their discount when spending trade).  The Chinese can be good as well, as they get some free culture at the start of the game to kick-start progress, though their real strength is if this gives them a great person that produces two culture.  The Germans and Russians are really not worth trying with (see this thread https://boardgamegeek.com/thread/626430/tracking-victory-types/page/1, which also illustrates the comparative difficulty of a culture victory to other victory types).
If playing as America, Rome, or Egypt, put your capital next to the incense on your starting tile; if you later get feudalism, any of your cities will be able to harvest this (note that it can still only be harvested by one city each turn, but it could be a city other than your capital if your capital can produce more than 7 culture through the culture action).
Wonders are key.  Stonehenge, The Louvre, and the Sydney Opera House are the big three you want to get because they give you free advances on the culture track.  The other nice thing about these is they add culture icons to your cities, letting you produce more culture when you hit that action.
Have at least one of your cities devoted to producing culture.  Early game, this will mean building culture producers in that city.  Once the city is ready, hit culture with it every turn.  Do not be afraid to spend trade to get buildings you need in that city (if you are not playing Egypt).  It is better to do this for a few turns if it can get you a solid culture producer faster.
Get technology that feeds your strategy.  Key technologies include: Theology (for the Cathedral and increased hand size), Metal Casting (for the culture production ability, with the added perk of giving you a unit upgrade), and Printing Press (for the University; you should pretty much ignore its coin ability unless you are pivoting to an economic victory).  Other technologies you should consider are Chivalry (for the culture production ability and the government), Monarchy (for the government), Mass Media (if other people have Civil Service), Pottery (for the hand size), Civil Service (for the hand size), and Philosophy (for the great people and the ability to build temples first and upgrade them to cathedrals later).
Cities that have poor culture production can still be used to produce large amounts of culture by producing incense.  Incense can turn into 5 or 7 culture a piece mid-game with the technologies mentioned above.  Don't be afraid to harvest incense before you have these technologies and hold on to it.  Don't use incense on any ability that produces less than 5 culture (don't use the ability on Currency and don't spend incense on abilities like Philosophy).  Go for villages, as they can help you get resources to fuel your progress.
The culture cards and great people you get help your progress.  Great people can help your culture producing cities produce more culture.  This is great (pun intended).  The culture cards can get you the resources to get more culture, can help defend against an aggressive military player, and can disrupt the production/buildings of someone going for an economic or military victory.  If you start advancing on the culture track early, this can start a positive feedback loop.  Also, if you have philosophy and extra non-incense resources, get some great people.  They can help start your culture production (especially if you get one of the great people with two culture icons).  Spending a culture card as an incense to get enough culture to buy another card may seem like a waste, but it's actually exactly what you want to be doing, since this pushes you both closer to victory and closer to the tier 3 culture cards, which are more powerful.
Favor archers when buying units, as you are definitely going to research Metal Casting, which puts those at level 3.
There isn't a necessary tier 4 technology for culture.  Mass Media can be okay, but really feel free to research anything as your first tier 4 tech if it helps you more.  Atomic Theory can be very powerful, as later culture cards (princely gift) give you access to uranium and taking two actions with all of your cities can let you get a truly silly amount of culture in a turn.  Alternatively, you could get one of the military techs to defend yourself.
Build cities next to desert rather than water; trading posts (which generate culture) are more useful to you than harbors (which don't).  Build later cities next to natural wonders if possible to get the extra culture.  The purple forest is probably the best.  Using scouts on natural wonders can be helpful too.
Build all three cities.  This is important in any strategy in Civ, and culture is no exception.
Don't neglect trade production.  You will need it to research technology and also get through the higher ranks of the culture track.
Have a government of Monarchy, Feudalism, or Democracy.  Monarchy gives you extra culture production, which is good.  Feudalism can let all your cities harvest incense if any of your cities has it, which can be used on resource abilities to make culture.  Democracy can help in that you get extra trade and give up something not particularly useful (you shouldn't really be attacking other cities if going for a cultural victory).
Don't neglect military but it shouldn't be a huge focus.  Military is for spare actions and spare technology researches.  The culture cards do help you militarily, and if you see a battle you can win, go for it, as this lets you steal culture or incense from your opponent.  But attacking their capital is not your goal.  If you are in a position where you think you can do it, it's probably because you have spent too much of the game focusing on military.  Make sure you defend your cities, as if they take one of your cities, they can steal one of your hard-earned culture cards or, worse yet, destroy a city you sunk lots of trade and great people you put in to developing.
Don't hoard culture.  Buy cards as soon as you can and play cards as soon as you can.  Obviously don't buy cards if it would put you over hand limit and you want all the cards in your hand, but holding culture is dangerous as it can be stolen by battles.
When you are getting to the end of the track, go as fast as you can.  Forget everything but culture and buy advancement on that track every single time you can.  Discard great culture cards if you have to; you're going for the win.