I think you would rather plug your UPS into the surge protector. I only use Tripp lite surge protectors as well. You could also do it the other way around and nothing bad will ever happen but I would rather protect the UPS from surges and then use the UPS to power your PC
Why would you want to do this? To get more power connections? That would put a higher load on the UPS than it was designed for so it won't work very well as a UPS. Any good UPS would include surge protection and they help to even out spikes/sags. I don't think any UPS model recommends using a surge protector with their models, and most building safety codes prohibit stringing two surge protectors together which is pretty much what you are doing. At many jobs I had, we would have inspections of the setups and they would make sure we did not have a bunch of things daisy chained. Of course in some training classes our facilities people stuck 3-4 of those things together, so really anything goes till you start a fire.