Coolant refilling problem!

I drained the old coolant (both the radiator & engine block), replaced the thermostat with a new one, and then tried to refill the system with a container of radiator cleaner and water, in order to flush the system.
I can add only about 2 quarts of fluid via the pressure cap opening on the overflow tank. (The specified cooling system capacity is 7 quarts.)
I ran the engine till the indicated normal operating temperature was reached on the gauge, but still could not add any more coolant to the system via the overflow tank
Ran the engine for a few more minutes to see if the fluid level in the overflow tank would drop but it didn't.
How do I purge or vent the air that is in the coolant system and add the remaining 5 quarts of coolant to the system?
I'm reluctant to run the engine to long with only a couple of quarts of coolant in the system.
It sure was a lot simpler when the pressure cap/filler outlet was positioned directly on top of the radiator!
HELP!