I also live in an apartment with no washer/dryer. I had planned on washing all my diapers by hand before I had DS. Here's what I did:
Purchased 4 dozen flats from Little Lion (LOVE these!), a few PUL wraps, and one wool wrap. (I found these weren't enough covers and got a few more after DS was born.) Thinking scrubbing by hand would be time-consuming and tiring, I got one of these:
http://www.laundry-alternative.com/washing.htm
I found I couldn't keep up with the wash (newborns go through a LOT of diapers, even if you don't change every time they pee, and DS likes to be held most of the day). So now DH takes the dipes to the laundromat once a week and I wash in the hand-cranked washer in between laundry trips when I can. (Warning: the crank broke off the washer after two months of moderate use, so now I have to rotate it with my hands instead. It's not that hard but some people might be annoyed at buying something that breaks. It was/is still worth it to me.)
Here's my system:
Rinse poop (DS has very sticky poop - partially formula-fed due to supply issues - so it does NOT rinse off well. If you exclusively breastfeed, I've heard it's easier). Toss these and merely wet diapers into dry pail. If I wash in the hand-cranked washer, I use the hottest water allowed by the manufacturer (194 degrees) and wash 6 diapers at a time with a tiny amount of All Free and Clear HE, do a warm rinse with vinegar, and then three warm rinses with clear water. Takes abot 10 minutes, not including the time to heat the water. I toss them on a drying rack in front of a fan and they dry in a few hours. I do this with wet diapers when I can and generally leave the dirty ones for the laundromat. If I have some stained diapers or ones that are still stinky after washing, I soak these in an Oxiclean Baby solution for a few hours, then wash in the hand-cranked washer.
Once a week DH goes to the laundromat, tosses the diapers in a couple top load washers. Does a cold cycle with no detergent, a hot cycle with an extra rinse, adding a quarter cap of All Free and Clear. I alternate having him add 1/4 scoop of Oxiclean Baby to the wash one week (to get rid of stubborn spots), and a 1/2 cup of vinegar to the first rinse the next week (neutralizes ammonia from urine). Dry on hot 1 hour. Yeah it's more money to do an extra wash but it gets them really clean. It's probably not really necessary.
If, like you, I decided not to do two wash cycles, I would presoak like PPs suggested, swish, and wring, then do a hot water wash. If I didn't have the option of a second rinse, I would rinse by hand after the wash, then wring and dry on the drying rack.
Hope that's not too many details for you. I'm pretty anal about some things and diaper washing is one of them!