My Macbook 13' won't turn on?

I just bought my Macbook 8 months ago. I turned it off last night and it was fully charged, unpluged the charger from the wall, and went to bed.

When I woke up this morning, I went to go turn it on again. When I turned it on, it indicate that there was 0% left battery power which seemed odd to me since I turned it off with a full charge.

Then it turned off by itself because I cause the battery power was THAT low, so I wasn't able to turn it off properly. I tried charging it and then seeing if it would turn back on, but that's not working either because I have a bad charger.

Could it be something to do with the power charger? This is already the second charger that I bought and it's starting to break on me the same way that the first one broke. Sometimes when it's in the light will indicate it's charging but the icon on the desktop will indicate that it's still on battery mode. I just ordered a new charger with a lifetime warranty.

I'm just worried that it's something worse then just a bad power charger! Any thoughts would be great, thanks.