You need:
required circuit gain,
frequency response,
supply voltage,
desired maximum current draw (power dissipation),
desired input impedance,
output load impedance,
operating temperature range.
From that you can determine the needed transistor parameters.
-Why the Hfe minimum?All we need is a supply voltage, what will be our load resistance and from what we have a input signal (signal resistance). And voltage gain.
And from BJT datasheet we need Hfe_min and sometimes Vbe and Hoe.
required circuit gain, I just need a colector current of 250ma, with the base voltage being 5voltsThe requirements I posted are the design requirements for the circuit you need to make. You can't do a design without first knowing what the requirements are, before you then calculate the circuit characteristics required to meet the design requirements.
You seem to want to put the cart before the horse and do the design before you have the requirements.