The pop explanation for acne is that ‘your hormones are going crazy’. The two classes of hormones best researched in regards to acne are the androgens and insulin/GH/IGF-1. There’s another hormone that I think is quite involved in acne but I’ll save it for a few weeks
Androgens include testosterone, dihydrotestosterone (DHT), DHEA and some of their derivatives. They are considered ‘male hormones’ because men have higher levels of them and they are responsible for many male secondary sex characteristics.
Testosterone can be converted to dihydrotestosterone by the enzyme 5α-reductase (5AR). There are two isozymes (multiple forms of enzymes) of 5AR: type 1 which is active in the sebaceous gland and type 2 which is active in the prostate gland. There is more activity of type 1 5AR (therefore more DHT) in sebaceous glands in the skin prone to acne (such as the face), which partially explains the acne in those areas. (DHT seems to promote acne more than testosterone) 
Androgens increase the size and secretion of sebaceous glands. DHT increases sebum production and the proliferation of sebocytes  and testosterone increases proliferation of sebocytes and works with PPARs to increase lipid synthesis . Androgen receptors on keratinocytes and sebocytes promote hyperkeratinisation, sebaceous gland development and sebogenesis (generation of sebum) 
Evidence supporting the role of androgens in acne:
- Elevated androgen levels due to androgen-producing tumours, administration of testosterone or DHEAS*, anabolic steroids or PCOS is associated with acne and people with acne have higher DHT 
- Severe acne is often associated with elevated serum androgen levels and females with severe acne have higher DHEAS 
- Those who lack functional androgen receptors and males who are castrated produce no/less sebum and don’t develop acne. Also a DHT inhibitor improves acne  
- Androgens seem to a major reason why acne usually
doesn’t occur prior to adolescence is more prevalent during adolescence. (Acne prior to adolescence is less
inflammatory as there’s insufficient sebum to support as much P. acnes (acne during this time is
associated with elevated DHEAS)) 
However, while serum androgens are elevated in severe acne the correlation doesn’t always apply to mild/moderate acne, If androgens are so closely associated with acne then males should have a much higher incidence and greater severity, but don’t and finally, some epidemiological studies find no relationship between androgen levels with the presence and severity of acne in men 
It’s unknown as to whether serum androgens or those made by the skin and sebaceous glands are responsible for acne, as sebaceous glands are capable of synthesising T and DHT from cholesterol and cholesterol from acetate. An explanation for the inconsistencies above could be that the skin and sebaceous androgens are more relevant or perhaps people prone to acne are more sensitive to androgens (have more receptors).
* DHEA sulphate (DHEAS) is an adrenal androgen and a metabolite of DHEA. DHEA is a precursor to testosterone and a weak androgen
Insulin, Growth Hormone and IGF-1
Insulin, growth hormone (GH) and IGF-1 have many roles in the body. Insulin is best known for increasing glucose transport into cells and GH/IGF-1 are best known as growth promoters.
Insulin increases IGF-1 and lowers IGF-1 binding proteins. GH is also known as somatotropin, and like other ‘tropins’ GH stimulates production of effector hormones, in this case IGF-1, which then carry out the biological effects.
Insulin and IGF-1 have other effects that can promote acne, such as stimulating lipogenesis of sebaceous glands, augmenting androgen receptor signalling and increasing free androgens by lowering SHBG (thereby increasing androgen activity). IGF-1 also increases DHEA sulphate (and is associated with DHEA sulphate levels), increases proliferation of sebocytes (and is associated with the amount of sebum excreted), promotes testosterone production in men and increases 5AR  
Evidence supporting the role of insulin/IGF-1 in acne:
- Acromegaly is a disorder of GH hypersecretion and is associated with elevated IGF-1, insulin resistance* and acne 
- Recombinant human IGF-1 increases androgens and acne, which normalises when the dose is reduced and when the treatment is interrupted 
- The number of total acne lesions, inflammatory lesions, serum levels of dihydrotestosterone (DHT) and dehydroepiandrosterone sulphate (DHEAS), each correlated with serum IGF-1 levels in women with acne 
- Retinoic acid (a metabolite of retinol/vitamin A) increases IGF binding protein-3, which inhibits the activity of free IGF-1 and is therapeutic for acne 
- PCOS associated with increased IGF-1 (2x higher), DHEA sulphate, IR, hyperinsulinemia and acne. Metformin is an insulin sensitiser used to treat PCOS and it reduces androgen levels and improves acne and PCOS symptoms 
- Acne is most prevalent during adolescence (mid
adolescence in particular) when GH and IGF-1 are highest 
High glycemic load diets are associated with hyperglycemia, reactive hyperinsulinemia and increased IGF-1. Low GL diets lower IGF-1 increase IGF-1 binding proteins and significantly improve acne after 12 weeks. 
There are associations between milk and acne (~1.2), especially skim milk (~1.4). High milk consumption is associated with a 10–20% increase in IGF-1 levels in adults and a 20–30% increase in children. This is because milk contains active IGF-1** and IGF-2 and dairy proteins (except cheese) are highly insulinogenic*** 
* GH may promote insulin resistance by increasing FFA
** There are high levels of IGF-1 in milk after pasteurization and homogenization. Bovine and human IGF-1 also have the same amino acid sequence
*** I’m surprised these numbers aren’t higher considering all the mechanisms where insulin and IGF-1 promote acne