Bleach is sodium hypochlorite and water with a small amount of residual sodium hydroxide from the manufacturing process. The sodium hydroxide causes the high pH but the sodium hypochlorite is more stable at a high pH which enables a longer shelf life. The sodium hypochlorite reacts with the compounds in a cell membrane causing them to break down disrupting the cell membrane and allowing the cell contents to leak (ie, it rips a hole in it thus killing it). The sodium hypochlorite reacts with organic compounds. Melanin is an organic molecule hence the whitening process.
That said bleach is a nasty product and has inherent risks in its use.
Tea tree oil and clove oil do have some anti-microbial properties.
^^ This is from DF. As mentioned, he is an Industrial Chemist with 20 years experience of formulating cleaning goods (as well as other products) and has made bleach in his current and previous roles.