2005 Dodge neon no crank no start

My mom has a 2005 dodge neon, at first it started dying awhile back so i replaced the alternator and it was running smoothly again then a couple months past and it started to misfire in cylinders 1 and 3 so i replaced the sparks plugs and with a jump it started (with no more misfire) so we got a new battery and it ran well for 2 days and died again so we figured it was the alternator again so i replaced it again and with a jump it started and died after 2 minutes so my current problem is it wont start without a jump and it wont stay running when it dies, when you turn the key over theirs a click in the fuse and relay box i have switched around the relays bought new fuses and same thing just a click when you turn the key so im thinking starter but even then that wont solve the problem of it dying after 2 minutes so im lost if anyone could give me suggestions i would greatly appreciate it sorry for long drawn out explanation

The battery has failed, replace the battery. If the engine is started with low battery voltage the idle air control motor will be out of sync, resulting in engine stalling and the battery voltage will be unstable resulting in stalling.