It all has to do with hormones. When your liver and bowels can't handle all the toxins and hormone fluxuations then you get acne via excess sebum, skin not flaking properly, non acidic skin so bacteria can grow.
It would take me forever to type how to eat properly and still maintain weight, clean your liver, bowels, etc... but I'll give you a short list of things that do this.
YOur diet is supposed to have an essential fatty acid ratio of about 2:1:1(omega 3 6 9 respectably). However the ratio is about 20:1 in favour of omega 6. WE meed way more omega 3 fats which are found in cold water fish, some wheats and plants but not much else. These omega 3 regulate prostaglandins(all 30) which in turn regulate hormones.
Now acne sufferers usually can convert these fats to long chains which are useful so it is best to take fish oils with lots of EPA's and DHA's.
You also need to take these with protectants so they don't get destroyed before use. Take this with lecithin, coconut oil. Other vitamins that protect the oils are E, cod liver(A and D), B-Complex.
YOu also need to clean your blood. Acne sufferes have dirty fats, protein anc carbs(especially proetin) in their bloodstream because they lacked enough enzymes to digest them . Make sure you take a suppliment with protese, lipase and amylase.
YOu can also take milk thistle, burdock root, and dandilion root to strengthen your liver(milk thistle is extremely good stuff and doctors use this for all kinds of liver problems, except oddly acne).
There's a lot more but if you do the above and simply wash your face with PH balanced soap thats antibacterial you'll be clear.
Oh ya, flucuations in blood sugar also add excess hormones. Try green tea(loose leaf) or some blood sugar regulator to keep it level.