starting problems

I have recently got a 1999 grand am 3.4 and i got in from work and it wouldnt start so i got a jump and i got home fine so i got a new battery and it was fine for about a month then today it wouldnt start again so i got a jump again and i got it home fine can anyone help

now when im driving the cd player lights blink with the turn signal also when i rev the car up all the light get brighter can my cd player will turn back on is it the alternator or something else

Some decent sentences would be nice.
I would do a VAT40 check on the alternator.... but thats just me.