Newborn pups need their mother for more than just food. Until puppies open their eyes -- at about 14 days -- and start moving around, they depend on Mom for everything, from food to warmth to help going to the bathroom. Sometimes mothers reject puppies who look too weak or appear sick, in which case bottle-feeding the pup is the only way he will survive.
Buy dried or canned puppy formula. This is sold through veterinarians and some specialty pet stores. Never feed a puppy cow's milk, as it can cause diarrhea and lead to dehydration and nutrient loss. You will also need a puppy nursing bottle and a container in which to mix the formula if you're buying powder.
Feed the puppy every three to four hours. This means throughout the night too. You should feed enough to make the belly round without making it feel too hard. When in doubt, it's better to feed small amounts more times than feeding large quantities, which can cause indigestion and even diarrhea. There's no specific quantity recommended -- just stop right before the puppy feels too full -- but as a general rule, a 10-ounce puppy should be eating about 75 cc of formula a day. If you're feeding about every four hours, that's about 12.5 cc every time.
Place the puppy face down, rather than on his back, when you're feeding him. This is the natural position in which he will drink from Mom, and it prevents him from swallowing too much air as he's drinking the formula. If you're refrigerating the formula in between feedings, you'll have to get it out of the fridge in advance, as the liquid should be at room temperature before you feed the pup.
Start the weaning process by the third week. You can slowly introduce wet dog food first. If the puppy takes to it eagerly, alternate between wet food and bottle-feeding throughout the day. Slowly reduce the amount of formula and increase the amount of dog food over the next couple of weeks. Most puppies are not able to switch to dry dog food until week 10 or so, when they have their teeth.