My 05 Dakota started and immediately shut off. Now it is acting like a dead battery but will not take a jump the code was 1113 and I cant find that anywhere. What could be the problem?

When you jump start are you putting earth to the body/chassis or to the battery??

how can you have a brand new battery and no power...
fix this problem first

You need to have the ECU checked for any codes.

starters, alternators and radios will drain the batteries the fastest

You have bad terminal..open the battry terminals pour some boiling water over them. then use a sand paper to clean then when ur done doing so install the terminal and put some grease ove them a bit not too much. and tight it well i think it will be ok then.

press and hold lock button on remote while pointing it at the senser.

Well if you cant jump start it thats a good sign that the battery is dead so replace it and you should be fine, and another way you can try is hook the battery up to a charger overnight and if it doesnt hold its charge then you will know if its dead or not.

When it dies, it won't start back up? Does it crank over after it dies?

It's 100% possible bud. I've actually had a battery blow out on me at 3 days old. I was just driving to the store and I shut the car off. I went and got some stuff and came back to a 100% dead car. It had a new altenator and the battery was all but three days old. It was so dead that even the overhead light wouldn't come on. I got a jump start from a friend and made it to napa. They tested the battery and said it reads perfect until you try to draw energy from it.

They told me it was a shorting cell, and that the battery was shot. It was replaced for free; as should yours if you still have the warranty information.

Well its because a mechanical problem. Look, sometimes a car reach to a mile when he got to be taken to a repair in a garage.