Battery light comes on and fades while on the gas?

Recently replaced the alternator when battery light came on and stayed on until it died while driving was checked and tested as a dead alternator.

Currently on start up car runs really rough with check engine light flashing until it smooths out and turns off and when driving the battery light will come on strong while on the gas and when I let off it will fade.

I'm not electric savy so I'm not really sure whenever to look and I don't want to throw a bunch of money at things that don't work please help.