How do you connect a long string of LED strips to power at both ends when including a controller?
Don't bother with the 120V system.  It's less safe and more expensive.  Learn to wield low voltage at long distances.  Feeder is the key.  

This is a 5050 system, which means it is very power-hungry. 

Because you're at 24V, you won't need to connect power to both ends of each 16.4' (5m) strip.  However, you will need to connect power to one end of each strip.  That means you will need to carry your 10AWG "feeder wire" along with the strips most or all of its way.

10 AWG wire sounds right, however it would need to be 10/3, assuming  your local Codes allow you to use the ground wire for DC return (a bold assumption). Remember you must use outdoor-rated cable, so UF for instance, never NM. 

If you want to control the two sections independently, then that is fine.   However, if your reason is because you are hitting current limits for 1 controller, couple things on that. 


The right way to handle that is actually 1 controller and amplifiers.  The controller's output is used either as power to some LEDs, or signal input to the amplifiers, or both at once is also fine. 
Amplifiers also let you use a different power supply for each amplifier. 
It is virtually impossible to keep 2 controllers in sync.  You must use the 1 controller+amplifier(s) method if you want them to match. 
Don't run this Cheese junk anywhere near its rating - it will fail if you do.  


That goes double for the power supply, by the way.  Don't run it at spec limits unless it says General Electric or Siemens.  I wouldn't even push a Meanwell near limits. For a cheap random Chinese thing with no UL listing, I'd derate it by about 66%. 

Mind you, all that "Sold by blah-blah and Fulfilled by Amazon" stuff evades all our product-safety laws, and has never seen the inside of a NRTL testing lab.  Since it's direct-ship (or drop-ship via Amazon), you're the importer for purpose of product-safety laws.  I don't particularly care about low-voltage stuff, but anything that touches mains electric ought to be UL-listed.