The best way to check your charging system is with a digital multi meter (<$30 @walmart/homedepot). If you're unsure of how to use one youtube is a great resource. Basically you'll need to recheck all the connections, both at the battery and the new alternator. Make sure everything is clean and tight. With the car sitting in your driveway not running on a new battery you should see 12.6-12.8v. Anything below 12.2 and your car won't start the first frosty morning this fall. With the car running and a new alternator installed you should see 13.8-14.4 v. Both measurements taken at the battery terminals.
If the battery when you first check it is less then 12.5, then the new alternator you installed needs to be checked. If with the car running you have less than 13.5 on the battery, check the bolt on back of the alternator where the battery cable connects. You can do this by just moving the red lead of the multi meter from the + battery terminal to the alternator bolt. If you are now reading 14v +- then you will need to double check the cable from that bolt to the battery terminal. It's either loose, corroded are a bad cable that needs replacement. If you still get 13.5v or less regardless of where you're checking than you'll need to cash in on the lifetime replacement of the "new" alternator you just installed.
These "new" alternators/starters/etc from parts stores are rebuilds and some of the rebuilders use cheap chinese junk parts that sometimes don't even last long enough to get the item out of the box. I had to replace a starter 3 times once till I finally gave up and bought a real "new" one.
Or you can have the charging system tested at autozone/napa/carquest hopefully the new employee that does the test knows what he/she is doing.