How do we measure one complete orbit? This is being asked from a lay-person point of view. However, please do not hesitate to give more detailed answers that require prior knowledge - if these answer the question more accurately.

My confusion currently comes from the fact that everything in the universe is moving in some way or another - so any reference point we use would also be moving/not completely reliable.

As such, how can we accurately determine that we have completed an exact 360 degree rotation around the sun (and as such, measure exactly how long it took)?

To be clear, this is asking about how we can determine that we have completed a single 360 orbit of our sun, and not the timing aspect of how we know how long a second is. If this situation is different for measuring the orbits of other bodies around their star - it would be great to expand on those differences as well.
When one asks the question, "How is a year defined?", there are actually a few different answers one can provide. What you're implicitly asking, even if you don't know, is how one defines a Sidereal Year. In effect, the Sidereal Year is the time necessary to complete one, full, 360 degree revolution around the Sun. This is opposed to the Tropical Year (or else the Solar Year) which is the time necessary to travel just slightly less than a full 360 degrees (due to things like precession and the definition of the Tropical Year).

So your question boils down to, how is a Sidereal year determined? In practice, this is often done by choosing a specific starting point and noting where the Sun is with respect to the stars in the Sky. Then, it will be an entire sidereal year (and a 360 degree revolution) before the Sun returns to the same position as previously observed. Once you've observed the Sun in the exact same position you know a full revolution has occurred of the Earth around the Sun. It should be noted that the motion of the stars over a year is almost completely inconsequential. The stars are so immensely far away compared to their own motion, that their apparent motion (otherwise known as the proper motion) is basically nothing over a year. In this way, we consider the stars "fixed".