Let me see if I can explain this in a way that is understandable. This all assumes you're having weather cold enough to set this up outside and it will freeze as carrying it outside after freezing doesn't seem practical to me.

You will need a bunch (depending on how large you make it) of small diameter sticks. Keeping it affordable, I would recommend those thin bamboo stakes they sell in the garden section to stake up your plants in the summer. Could go with dowels but it will cost you lots more. Puncture the cups on each side at about the level you plan to fill it with water to. Feed the cups, one at a time, onto the stick, preferably inserting the narrow end of the stick first as this should give the tightest fit of the hole in the cups around the stick which will be important in step 2. These sticks are maybe 3' long so if you want more, put second hole in end cup, next to first one, so you can start a new stick.

Set your cups on a stick where you want them/where they will be frozen, fill with water up to the level of the stick. (with luck, expansion will push the freezing water up around the stick) After this is frozen, set your next row of cups on a stick on top. Since there may be some leakage at the stick, if you are really committed, wait until late, sundown or later to do this so it will freeze overnight. Pour water into the next row of cups, feel free to fill higher than the stick, a little leakage won't hurt. Then pour additional water down the side of the bottom cups to top up and provide for the freezing together that you hope to accomplish.

In theory, you could branch this out to the sides as well, using the same stick technique, adding stability and allowing for building a 3 dimensional structure. Not a bad idea as a well placed snowball on a one dimensional structure could be catastrophic. Hope all that makes sense. I can visualize it in my head but explaining is not as easy. Good luck, sounds like an entertaining way to play outside. This was my last snow game creation. Can't do this now, Raynaud's syndrome has gotten bad, can't handle the cold at all.