This will get you count:
get-alias | measure
You can work with the result as with object:
$m = get-alias | measure
$m.Count
And if you would like to have aliases in some variable also, you can use Tee-Object:
$m = get-alias | tee -Variable aliases | measure
$m.Count
$aliases
Some more info on Measure-Object cmdlet is on Technet.
Do not confuse it with Measure-Command cmdlet which is for time measuring. (again on Technet)