In order to successfully harvest yeast from a bottle conditioned beer, the yeast must be viable, and the yeast must not be contaminated. That seems like a no brainier, but it is important to consider. If you have a microscope I would check before spending much time with the yeast. If you don't have a microscope I highly recommend that you start by plating the dregs. All of the dregs that I have looked at contained at least some bacteria or mold, and in many cases there have been enough to dominate the culture.
One good thing about yeast is that it just does it's own thing. If you want you can check it periodically, but if you forget about it for a few days it will just keep going on it's own. The only real cost is a little time and some dry malt extract. Using this method will take five short interactions over a couple of weeks, and 200 grams of DME.
The number of viable cells can vary quite a bit from one bottle to the next. In my home brews the cell count ranges from about 300 million to 2 billion cells suspension in each bottle. The target number of viable cells used by some brewers is 1 million per ml which is about 350 million cells. (1) Although a beer I recently harvested from had only 7 million viable cells. When working with a bottle with very few viable cells it could take a week to see activity even when starting with a few milliliters of wort!
To keep the numbers round lets just use 1 billion total cells. The viability will likely not be great, so lets assume 50%. That's 500 million cells. The set of experiments done at White Labs that were used to develop the Mr. Malty calculator. (And the same data that was taken to produce Yeast Calc) Indicate that the most new cells per volume can be produced with an inoculation rate of 65 million cells per ml.
2 billion cells in 30 ml yields a pitch rate of 67 million per ml.
To make a 1.036 gravity wort 3 grams of DME can be added to 30 ml of water. Fermentation of this wort will yield about 3.7 billion cells. For the next step, now that we have healthy yeast cells, we can make a larger step. 300ml is conveniently to the bottom of the neck of the 12 oz bottle. 30g of DME should be used to make the 1.036 wort. This will yielding 15 billion cells, and then you could go as high as 2 liters yielding 70 billion cells(1/2 gallon of water with 1 cup of DME) , but two steps of 750ml and 1 liter will yield 100 billion cells.
Of course you will want to keep things as sterile as possible during this processes. A foil cap on the bottle is probably a good idea. Also, once the bottle has been opened it is ripe for infection so you want to start this processes the same day if possible.
If you wanted to have the yeast ready as soon as possible the minimum time line is just over a week, but if you are a busy person, you could stretch the time line out to a month without any real consequence.
Here's how to do it:
1) Add to the beer bottle containing the yeast slurry 3g of DME and 30ml of water. Let it ferment for four days. (It can take a while for yeast to become active after the long period of dormancy.)
2) Fill the beer bottle to the start of the neck with water and 30g of DME. Let it ferment for two days.
3) shake the bottle to suspend all the yeast and pour it into a 2 litter or larger container. Add 75 grams of DME and fill to 750ml. Let it ferment for two days.
4) add 100g of DME and fill to 1 liter of water. Let it ferment for two days.
5) crash, decant, and store your 100 billion cells of yeast.
(1) Brew Like a Monk