For a modest system that is focused on SQ, you do not need a new alt. You do not need to add batteries. You do not need a kW or more on the subs.
Decide what features you want in the HU and pick one. Unless you want Bluetooth and nav and so on, that should run you around $3-400 for a good one. Next pick your front speakers. Don't worry about rears, you can add them later if you decide that you need them. SSA is selling Bravox two-way components for $195 and lower. They get great reviews and that is a great price. Decide if you want 1 sub or two based on how much space you're willing to use. I'd stick with lower powered subs so you're not losing out on efficiency by trying to move a huge coil and stiff suspension. Don't get me wrong, some of the higher power handling subs will sound great on a really small amount of power but in a sealed box you'd give up a LOT of output with those. With your budget I'd suggest a single 12. I know that you said you liked 10s but there's nothing that a 10 can do that a 12 can't and the 12 will give you more output with the same power and a single 12 takes up less space than a pair of 10s. Finally decide how to power it all. 50-100w is plenty for the front stage and 300-500 is plenty for the sub. Something like a Sundown 100.4 would be able to run he whole system and make for simple and inexpensive wiring. Single 4ga run from the battery to the amp. Budget $100 or so for installation stuff like wire (power and speaker) and connectors and you're there.