You'll need the full power of an electricl outlet to do this. At least 1500 watts or 10-12amps. So you don't have power?
You can use a 12v pump and a solar panel. Look at Harbor Freight 12v pump and look for Harbor Freight 60w solar panel. If you don't have power, you'll need to pump water up and use gravity feed to get pressure. A rain barrel about 30ft higher then where you want water should give you enough water pressure. This a float valve helps: https://www.amazon.com/Kerick-Valve-MA052-Float-Adjustable/dp/B0077RAP1I/ref=sr_1_1?ie=UTF8&qid=1465850762&sr=8-1&keywords=float+valve
i have something like this in mine to fill the tank automatically. might need to hunt around for the right config but if you are gonna be that lazy (which is awesome), go all the way!
I'm not sure I totally understand what you are trying to accomplish. But, if you want your bucket to have a consistent level of water inside, then use this - https://www.amazon.com/gp/product/B0077RAP1I . I use them on my sub-irrigated planter system to keep the planters supplied with water at a certain level. I hook them up to my irrigation system though you could just keep them supplied with pressurized water constantly. This is basically the same as a toilet tank valve.
This style of "float valve" is very common in many commercial systems. Conceptually it is no different from how your toilet refills: https://www.amazon.com/Kerick-Valve-MA052-Float-Adjustable/dp/B0077RAP1I/ref=asc_df_B0077RAP1I/
I would suggest sourcing a "female NPT to barb" adapter that is the compatible -- a local ACE could probably help you out. You will then connect soft poly tubing to the barbed end of the adapter using a hose clamp. The poly tube will need to be fed from a bucket of the water you want to use to refill the kratky jar. The water level in the "makeup" water will always need to be higher than the high point in the kratky jar to pressurize your line. Know that if you are keeping a nutrient solution in a single container for long periods of time, you may need to stir them as salts can settle. If you want to be really fancy, go for a magnetic stirrer.
there is a washer between the inside and outside the drilled hole in the plastic container.
I have done very similar stuff! Ill take some pictures for you to explain how it all works.
I was doing some thinking about your potential setup and i realized that if that unit's tank is plumbed up to drain in to a 5gallon bucked, there's nothing to prevent the bucket from overflowing. I have a solution for that though! all it takes is for the drain pipe to run through a float valve to the bucket.
this is the float valve, which is 8$
https://www.amazon.com/Kerick-Valve-MA052-Float-Adjustable/dp/B0077RAP1I/ref=sr_1_6?keywords=float+valve&qid=1565294395&s=gateway&sr=8-6
what that would do is stop the flow of water in to the bucket so it doesn't overflow. The hose will then start to fill up until it backflows into the unit's reservoir, and because the hole you drilled is below the max fill line, the unit will fill up and then shut off, at which point you will have to dump the bucket and the units reservoir, but should increase your capacity by about 3-4 gallons, depending on how high up the bucket you put the float valve.
I know it all sounds a little complicated but its really quite simple once you understand the parts.
check out this awesome video explaining the part i just talked about. Imagine his bin is your bucket(you could also use a tote bin like that) and imagine that blue tube is connected to your unit's tank.
https://www.youtube.com/watch?v=Cm-DIHR6uVA
You could use a pretty standard mechanical float valve to refill water reservoirs. Something along these lines