How do we write damping factor for a general RLC circuit
For a general RLC circuit -
1.Indentify the output  asked(given) in question !
2.Find transfer function i.e $$T(s)=Vout(s)/Vin(s)$$ using KVL, KCL , Mesh , nodal etc. Methods
3.And once you get the Transfer function(which will be 2nd order most likely ) then convert that Transfer function into standard form i.e
$$
\mathrm{
H(s)=K \frac{\omega_n^2}{s^2 + (2\zeta\omega_n)s + \omega_n^2}
}
$$
4.And after comparing with standard form you'll find $$\zeta$$ which is damping factor
4.without prior knowledge of output, transfer function of any system cannot be obtained and hence asking for damping factor without output doesn't make sense !