Steve,
You need to look at Ohm's law. Voltage = Watts X Resistance. Or Watts = Resistance / Voltage. You already know you don't need an alternator to run the LED lights. They run off of DC voltage and your battery is DC. Your charge controller is there to make sure you don't damage the battery. A car battery has smooth cathodes inside which give it a higher "cold cranking amperage". That's because a car battery needs to turn a lot of metal in the car before the engine starts up and makes the alternator run to recharge the battery. A deep cycle marine battery has perforations in the cathodes, which give it more surface area. It's not able to handle the work that a car battery has to put up with every day, but can handle frequent charging and discharging.
Once you know how much "work" your led system needs to perform, i.e. the number of Amps, pick a battery that can handle those Amps. Look at the Watts that the lights are going to need to work. Once you have the numbers for Amps and Watts, you can figure out how much of a solar panel you will need to produce those Volts.
You can probably do with one good solar panel from Harbor Freight, but you might need more than one marine battery. Don't forget about cloudy days. Solar panels will produce electricity even in low light, but if you're having bad weather for a week or so, you'll need more storage capacity to keep the lights on.
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