The paper studies the dynamics of a Duffing oscillator with time-delayed feedback. The delay differential equation (DDE) describing this system reads \[ \ddot x(t)+2\zeta\omega_0\dot x(t)+\omega_0^2 x(t)+\mu \omega_0^2 x^3(t)= u\omega_0^2x(t-\tau)+v \omega_0\dot x(t-\tau)\text{ for }\zeta>0\text{ and }\omega_0>0. \] First, the authors locate the Hopf bifurcations in the parameter space, also expanding the frequency and the direction of the periodic solution branch emanating from the Hopf bifurcation in a multiple time scales analysis. Second, periodic solutions near the Hopf bifurcations are computed (regardless of their stability) using a single-shooting method matching the initial value and its image under one Poincaré return map on an equidistant grid. It should be mentioned that there exist already alternative powerful numerical tools, based on collocation, for the continuation of stable and unstable periodic orbits [D. Roose, T. Luzyanina, K. Engelborghs and W. Michiels in: Silviu-Iulian Niculescu (ed.) et al., Advances in time-delay systems. Berlin: Springer Lecture Notes in Computational Science and Engineering (2004; Zbl 1065.34077)].