Yes, PVC's are or can be in some people a Hypo symptom. That goes away with thyroid. Typically tachicardia (rapid heart beat) is more associated with being Hyper.
So I would agree with Goolarra. Take care of the thyroid first while keeping careful watch of the palps and need if necessary for beta blocker. I would also recommend that you start very small and work up in very small increases.
I have Wolff-Parkinson-White syndrome. It causes tachycardia and palpitations. It was completely under control without meds until I started thyroid medication. My first year on thyroid meds was difficult, but since I've gotten my meds adjusted properly and been on a stable dose for a long time, my WPW is the best it's been since I was a kid.
It takes some getting used to, and I now take a beta blocker. In my opinion, getting your thyroid hormones right, whether they're too low or too high, is the best thing you can do for heart arrhythmias.
As you said, you never had this problem before your thyroid hormones got too low.
You want to start out on a low dose of meds (around 25 mcg) retest frequently (every 4-5 weeks) and increase slowly and as tolerated. I was started on much too high a dose, and that can exacerbate arrhythmia.