First things first, I'm an Electrical Engineer, I've been doing car audio for 12 years now, and I can tell you that without a doubt that capacitors are the biggest scam in the history of car audio.
Your alternator went out because of your subs, that's the simple truth. Most vehicle only have about 20 to 30 amps of overhead on the alternator, which can easily be eatin up by a pair of subs, so mix that in with an alternator that was probably already on it's way out, and that's why it failed. There are ways to keep this from happening. A kinetic battery is an ok idea, but it's really only a bandaid for a much larger problem.
The easiest way to describe this is in terms of water, if a river is the an analogy for the power being supplied by the alternator, then a lake is the battery. If the river is running slower than the dam on the lake is letting water out, then the lake will eventually go dry, regardless of how large the lake is. A capacitor is like having an even smaller lake, but runs into the same issue, if there's more water leaving than there is coming in, then that lake will still go dry.
So a capacitor is a waste of money (with the exception of systems that have exceedingly supplemental power, in which case they are used for to keep signals passed between amps continuous rather than the actual "power flow"). A better battery isn't a horrible idea, but it doesn't actually solve any problems. I would recommend either buying a high output alternator, or trying to find a much smaller sub that only runs on 200w or so.
As for everything else, most of the people are correct on the brand recommendations. Dual is an absolutely horrible brand (sold them for 4 years and never once had one last more than a few years). Sony speakers arn't horrible, but there are definitely better options. If you are looking for a deal though, you are actually in luck, all the BestBuys nation wide are in the middle of their annual car audio transition, so all of their decks, and most of their speakers are on clearance prices. Plus, they're discontinuing Rockford Fosgate, so all their RF stuff is CHEAP!! I was in the other day and picked up the 100.1db amp (normally $700) for $200 brand new. I know they have a few good Pioneer decks (DEH-P5100UB) for about $150, which is a really good buy if you are looking to build a system, since it has very easy to use controls, excellent crossover capabilities, and a clean looking OLED display. As for speakers, they have some nice RF speakers on sale right now, so go take a look!!!
- To recap though, don't listen to the guys tellin you to buy a cap, they don't know what they're talking about. Save your $100 bucks and invest it in a better alternator or better stereo equipment. Check out BestBuy, but hurry, it won't last forever.