First off...bala's are not good for 10 gallon tanks i would take them out asap.
I know exactly why your fish are dyeing. You havnt cycled your tank. Do you know what cycling the tank is? Well...your obviously going to have to be cycling the tank with fish ! Lol. here is what it is in short terms for ya.
When the fish poop that turns into AMMONIA which is very bad for the fish and is what killed your fish. Over a period of time bacteria will form in your tank from the AMMONIA and the bacteria will eat the AMMONIA, than the AMMONIA will turn into NITRITE'S, which is very toxic to the fish also, and than when there is enough bacteria in the tank it eats the nitrite's and than turn's into NITRATE. Which is not hardly at all toxic to the fish.
Do you have a test kit? I would recommend buying a MASTER TEST KIT. You will need at least the kits for ammonia, nitrite, and nitrate. I would get that ASAP and test your water.
I'm pretty positive this is called the " New Tank Syndrome". I did it too when I first started!
Your water when finished cycled should be this:
AMMONIA : 0
NITRATE: under 40
also, neon tetra's are not good hardy fish for the cycling process because the ammonia kills them easily
I would recommend taking all of your fish back for some credit or something and return them for lets say a few zebra danios or platy's...molly's. Those are just a few hardy one's that i can think of.
Hope this helped and If you have any more questions about this than us at fishforum will be sure to help you!!!