Battery? Malibu 2005

Beginning of January when we had negative temps
my car would not start I forgot to deal with it [was
a Saturday] on Monday couple days later I went to
go charge it.  But it started right up with no
problem. A week later i went to drive it and the
lights were on radio was working everything but it
made a clicking noise when trying to start. I figured
I would put the battery charger on about 3 hours
later it started.  No problems for 3 or so weeks and
I started it was running for about 20 minutes [530
a.m. had to make sure I could see]. Drove about 1
mile maybe 1.5 miles and all of a sudden interior
lights dimmed car slowed  [thought I was running
out of gas] I pulled off good as possible [of course
snow storm] lights stayed on I decided to see
about starting it and it sorta cranked over but didn't
stay running. Put my flashers on radio was off
headlights were off [by choice] about 30 minutes
later everything died no flashers lights nothing.
[Honestly don't know if my headlights dimmed
while driving I was paniking and could barely see
with the snow as it was]. Towed it back put on
charger and few hours later it started. Few days
later started again.  We are going to look at it this
weekend and was wondering if anyone had any
ideas on how to figure it out if it is starting again.
Things people tell me could be:
Battery
Alternator
Battery cables bad connection

Thank you!
Battery sticker says 1/11
Has over 301k miles
Don't know much about the history I do know at
first years ago when we came into possession of it
it would lock/unlock multiple times craziest thing.
But hasn't in over a year.