Q:

Strategy to win the Khan scenario? 

How do you approach the Khan scenario that comes with the basic Civ-5? 
Couple of ideas I'm working on, but always nice to hear from those who have tried and succeeded...

Send horsemen off in different directions to explore the map whilst growing/building new cities? 
After about 40 turns i figured that was going nowhere fast... but I did learn the map so I have unavoidable knowledge of the terrain next time through. 

Send horsemen to conquer the nearest city-states, before turning attention to other civs? 

Strike at the biggest rival first and the rest will fall with ease? 

Strike at the smallest rival first and work your way up? 

I'm down at the middling levels of skill.  Does your strategy change as you make it harder?  How/Why? 

A:

Unless you start conquering quickly, you won't be able to conquer enough civilizations in time.  My tactic (in difficulty level 5 if I remember correctly) was to immediately use my horseman to conquer the nearby city-state (to the south east) and then the Siamese civilization.  It's not an easy war to wage but conquering these cities is a pretty fast way to expand your borders quickly. 
Use the terrain to your advantage, as well as the powerful Khan generals and the ranged knights.  Avoid fighting in hills until you can get a few pikemen. 
I continued to conquer and advance to the south east until I couldn't (that's Siam + China), and then turned my attention west and attacked India and Persia, ending the game on time. 
My biggest problem throughout this playthrough was maintaining happiness.  Try to prioritize conquering cities with resources and burn down cities that don't really help.  The bonuses from city-states are also very nice, so it's recommended you take them as soon as you can afford to. 

A:

This strategy was quite successful at Prince level after a few failed starts:
Ignore the city-states
Unlike regular civ games most of the city states don't have a luxury resource so don't be thinking they are happiness neutral.  I found them little net benefit as they cost time and happiness.   Nonetheless they are cheap - given the combat bonus against them - and they do return a bonus of units, money, a technology, or a culture.   But I had better success if I ignored them altogether.  Their units will mostly leave you alone.  Make peace with them when not blocked by an alliance and take out any barbarians they may be moaning about, as convenient, might make some like you by the end. 
Conquer the next nearest civilization thus...
Don't bother waiting to declare war till you have a big build-up, or even know where their cities are.  Speed is important and, if you're lucky they may send out their forces to meet you en-route which is to your advantage as outside their own turf they get less healing, less city cover, and you have more room to manouveur.  Try not to war with more than one at a time of course. 
I found four keshiks the minimum necessary as one is generally out of action recovering or clearing up some miscellanous barbarian or whatever and you need at least three to subdue a city inside two or three turns.   Keep one general with the unit, on point, and expend the spawned generals as golden ages. 
The keshiks will conquer anything you need, just be careful to keep them alive and strong by always moving them out of harms way before the end of the move.  So step in, fire, step away.  Thus they accrue many upgrades.   However they cannot actually capture a city so always keep a melee unit or two (eg.  horsemen) in the rear to make the final blow then recover and guard the burning city whilst the rest of the army moves on. 
Raze the cities as you go.   It cures the happiness problem.  There is some discontent but it dissipates as the city burns down and you can proceed to conquer cities without having to worry about an endlessly unhappy populace. 
Leave their capital till last as it seems to get some bonus over regular cities and is typically the hardest to defeat.  Start with the nearest or smallest city you see.   You can't burn their capital so reserve 600 coins from the spoils, puppet it until the revolt ends, then occupy and immediately spend that money to construct a courthouse.  This counters the unhappiness problem. 
By the time you get to the last two civilizations your army will be quite a long way from home, so take care to keep your army alive and occasionally dispatch a new unit from home to make the long journey to join them; thus accommodating your occasional mistakes or speed up the conquests.   Keep an eye on the time as you only have 20-25 turns per civ, so not more than 5 turns to find and destroy a city is the pace you must maintain. 
For culture, finish off the militaristic one thence it doesn't matter a lot. 
For technology, it doesn't matter a lot either.  Perhaps push for an armoury as one of those might save a tight end-game.  I didn't bother with sailing and stuck to land based conquests.  You don't have time to research, build, and deploy advanced units like gunpowder and cannon, although sword and longswords can also be helpful to have around if the endgame is tight, but it seems unlikely to change the result.  Perhaps you'll run a tech focused strategy and post the results for us? 

A:

Keshiks are the key to your victory.  They can shoot and move to a safe distance.  After four promotions you can shoot twice so you get promotions very fast.  A task force of 2 keshiks with 4 promotions, one horseman, and one general is unstoppable.  The keshiks bring down the defenses, the horseman attacks and recovers in the burning city. 
I won the scenario using the following strategy:

I upgraded one horseman to keshik on the first turn. 
On the second turn, I started war on the closest city state.  I first killed their units, then took down the city. 
Then I attacked Siam, afterwards China, then India and Japan in parallel. 

I finished with 2 turns to spare mostly because I lost some advanced keshiks due to not paying attention to terrain features.  During the game I purchased most of the new offensive keshiks in my capital, but only after having an armory built so I could get the out with 2 promotions.