How to unlock the factory dungeons on cookie clicker

The answer to your question

The base amount of milk is 0%. The current formula for calculating milk is milk = achievements / 25. That means that for every achievement, you get 4% milk.

Are you sure 58 is the number shown in the stats page? I currently have 40 achievements, giving me 160% milk (as expected).

I have just discovered that not all achievements count. The August 27th update states:

due to popular demand, retired 5 achievements (the "reset your game" and "cheat" ones); they can still be unlocked, but do not count toward your total anymore.

If you hover over one of these achievements, it'll say [Shadow] in the tooltip. An example of this is:

Uncanny clicker ("Beat the clicking speed world record."):

I got this through using MouseKeys to sort of gain an advantage :P

This means that even though it's displayed in the stats page, it doesn't actually count when calculating milk %. Thus, even though I have 42 achievements, since one of them is a [Shadow] achievement, the game calculates my milk as 41 / 25 = 164%.

More milk facts!

The text next to milk is just flavour text. When milk is less than 100%, it's simply plain milk. When it's at least 100% but less than 200%, it's chocolate milk. Otherwise, if it's at least 200%, it's raspberry juice.

Milk will only affect Cookies Per Second (CpS) if you have at least one kitten upgrade. The first one, Kitten helpers, is unlocked at 50% milk and adds a 0.05 * milk multiplier. (For instance, at 50% milk, your CpS will be multiplied by 1.025.)

The other upgrades are:

At 100% milk, Kitten workers. This adds an additional multiplier of 0.1 * milk.
At 200% milk, Kitten engineers. This adds an additional multiplier of 0.2 * milk.

Thus, at 200% milk with all Kitten upgrades, your CpS is multiplied by 1.1, then 1.2, then 1.4, for a total multiplier of 1.848.

Note: sometimes, for whatever reason, getting a new achievement while owning at least one Kitten upgrade will not change your CpS. To force the game to recalculate your CpS, go to your address bar and type in javascript:Game.CalculateGains() or javascript:Game.recalculateGains=1.

^{Interestingly, a comment in the code says it should be 1% per achievement and that raspberry juice should be strawberry juice...}

You need to have Cookie Clicker as your active tab to let Raptr track your gametime and the browser doesn't need to have focus. If you're browsing with multiple tabs, everytime you switch out from Cookie Clicker, Raptr will notify you something like 'Cookie Clicker - 9 seconds played'.

This behavior has been tested with Chrome and Firefox. With Explorer I'm under the impression that gametime isn't tracked at all.