2009 300 will not start
 

Hey all,
I have a 2009 Chrysler 300 that is giving me issues with starting. First, I want to say that I just took it to the shop today to get the starter and alternator checked after I had been told the battery was fine. The mechanic told me that it was the battery so I had that replaced. I got in it several hours later and it started fine without a problem. I drove it maybe 5 miles then turned it off for maybe 10 minutes while picking up my kid. After attempting to start it a couple times (It was like I had left the lights on and drained the battery down) I got it to start. Now, I am at home and when I attempt to start it, it is starting but it is doing so like the battery is weak in it. Can anyone give me an idea what this might be before I end up taking it back to a mechanic. If it is something I can do easily at home I would rather not pay someone to do it.

Note: The first time the battery was checked I was at a parts store not a mechanic. The mechanic checked it the second time.