Today, our 'horizon' is the cosmic background radiation 'wall' which exists at about 300,000 years after the Big Bang. With 'neutrino telescopes' we may eventually be able to detect the neutrino background from the Big Bang and this will take us to a fraction of a second after the Big Bang. That's about as close to the Big Bang we will ever get because the technology will be very expensive to develop. As for seeing ourselves, no, we will never be able to do that. We may be able to see galaxies of the same type as the Milky Way being born, but that would only give us a statistical picture of our origins, not an actual image of our particular galaxy.