I would be very careful about using bleach on skin.
Bleach damages tissue and can make it more susceptible to disease. For example, washing syringes used by IV drug users as was suggested for many years with bleach makes one more susceptible to HIV (plain soapy water works better).
It is well known that bleach can damage skin (see MSDS). In the event of "serios skin contact" with bleach it is recommended that you wash with a disinfectant soap and use an anti-bacterial cream. In other words, exposure to bleach can make you more prone to infection. 10:1 dilutions or 100:1 might be better tolerated.
By the same token, Nonoxynol-9 was shown to kill HIV in-vitro but in-vivo it was found it could do more harm than good due to tissue damage - after years of use by consumers.
Alcohols and hydrogen peroxide may cause tissue damage in some cases, particularly on mucous membranes. Alcohol gels at least contain things to make them easier on the skin and appear to be ok to use.
Sterile Manuka honey looks worth checking out. Wounds reportedly heal in about half the time compared to most modern treatments and it has been shown to be helpful on MRSA. Some pre-prepared manuka honey dressings have been FDA approved (but are expensive). But beware of diluted preparations. Pure sterile manuka honey can be imported in 500g jars and 25g-80g tubes of sterile manuka honey can be purchased in the US. Not all varieties of Manuka honey contain the same amount (or any significant amount) of the active ingredient that makes manuka honey more effective - it depends on the variety of manuka the bees feed on. Even plain honey has a very long history (8000 years) of being used as a wound dressing, though largely forgotten today with the aggressive marketing of commercial products. However, non-sterile honey may contain Closridia or other disease causing factors.
Tea tree oil is known to kill MRSA but if it is diluted too much it not only doesn't kill but breeds tea tree and antibiotic resistant strains quickly.
Dilute Benzalkonium chloride also breeds resistant strains. I have heard triclosan has the same problem but the research on that appears a little questionable.
Oregano oil appears to be a potent MRSA killer.
From personal experience, a water based gel lubricant (KY jelly/Surgilube/etc) is very helpful for fungal infections (better than more expensive anti-fungals). It helps moisturize (without promoting growth due to moisture) and heal the skin and the preservative used, chlorhexidine gluconate, is effective in very low concentrations against candida. However, the concentration is much lower than in chlorhexidine gluconate surgical scrubs such as hibiclens. Thus the suggestion to use tea tree oil in KY seems promising, though the tea tree oil concentration should be at least 5% (but not too much higher). Adding neem oil and oregano oil would be worth investigating. But the combinations haven't been clinically tested.
Strong disinfectants may do more harm than good due to tissue damage. Diluted disinfectants breed resistance. Whatever doesn't kill the bugs makes them stronger.
When cleaning surfaces, reuse of a wipe on more than one surface, and use of non-disposable mops, rags, etc. can spread MRSA.
I don't have MRSA (I hope) but my housemate is suffering from a post-operative MRSA infection.
Bleach? That's the craziest advice I have ever heard. I hope nobody takes this advice!