How to determine whether a organic compound is diamagnetic or not?
If all the electrons are spin paired then the molecule will be in a singlet state$^*$ and will be diamagnetic, if they are not so paired then it will be  will be paramagnetic. The rules are the same whether or not the molecule is 'organic'.

Most 'normal' organic compounds, e.g. methane, benzene, etc. have paired electrons in each of the orbitals from the lowest one to the highest occupied one (HOMO), and thus are diamagnetic. The orbitals including the LUMO and up are unoccupied.

Radicals such as $\ce{CH3\cdot}$ have an unpaired electron (doublet state) and so are paramagnetic. 

If a biradical is formed e.g. carbenes (see IUPAC Gold Book 'biradical' for an example structure) then it will depend upon whether the spins are paired in a singlet state (diamagnetic) or parallel as in a triplet state which will be paramagnetic. Often singlet and triplets are close in energy with the triplet being lowest. Similarly, dioxygen has two unpaired electrons and forms a triplet ground state so is paramagnetic. Nitric oxide  has one unpaired electron and is paramagnetic.

If a molecule is put into an electronically excited state by absorbing a photon then both singlet and triplet states can be produced, however, theses are short lived and decay back to the ground state.

$^*$(singlet /doublet/triplet refer to spin multiplicity, spin quantum number  $\pm 1/2$ and total spin e.g. $S=s_1+s_2$ gives multiplicity g  as $g=2S+1$)

example of biradical from IUPAC Gold Book.