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The laptop won't be using the full 4.7A all the time, that's the max output when the computer is running hard, and the internal battery is being charged at the same time. You should find that the lappy only needs between 1.5 - 2.5 amps when it's running.
I bought a deep cycle 200Ah 12v AGM battery a few weeks ago and that sort of thing is definitely NOT portable, it weighs 62kg and is BIG.
Definitely get a deep cycle sealed (either gel or AGM) rather than a regular car battery or float type gel, and you really don't want to discharge to less than 30% capacity. A 100Ah should do the job fine, as long as you charge it properly. (Correct charging is a whole other world of complication!)
Cheers,
Jason.
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