Oxygens usually have an oxidation state of -2, except in the case of peroxides. So you can basically do oxidation states this way: Assume all bonds are completely ionic and have no covalent character. The atom that is more electronegative gets both electrons from the bond and thus oxidizes the other atom by one electron. So for atom B above, there are two bonds to H, one to C, and one to O. Hydrogen is a bit less electronegative than carbon so carbon gets the electrons. This makes the hydrogens both +1 and the carbon now -2 because it's taken on two electrons. That carbon is also bonded to another carbon. Since the electronegativity of carbons is the same, we ignore this bond. Go to oxygen. There's one bond to oxygen, which is more electronegative than carbon. Thus, oxygen gets the electrons in the C-O bond and oxidizes that carbon by one electron, making it -1 now.
Alternatively, you could just tally up the oxidation states of the atoms around the atom of interest and assign that number with the opposite sign to the atom of interest. In that case, oxygen is always -2 except in peroxides and so the -2 is shared between the two carbons it's bonded to. So each one "sees" -1 charge. Atom B also is bound to two hydrogens and a carbon. Hydrogen has a +1 oxidation state and C-C bonds don't count because they're the same atom. So around atom B, you have -1 + (+1) + (+1), giving you +1 charge around that carbon. Thus, that carbon has a -1 oxidation state to balance everything out.
