Are you working in a group? In my opinion, building a liquid fuel rocket is a bit much for a single person.
Here is a static test fire of a liquid engine built by a college group that i am involved in.
https://www.youtube.com/watch?v=U6w6LnCI5zQ
As you can see, this was run without a cooling jacket, which turned the copper nozzle throat red hot. So a cooling jacket was made, and will pump in either the fuel or oxidizer (i forget which - i havent looked at it for a while) around the nozzle. Coolant will work for static test, but probably not practical for flying, if that is your eventual goal.
Some other final observations:
- Cost: This rocket motor that we made cost in excess of $20,000. Unknowingly how much more without the access to the tools/machines/facilities already in place
- The bolt hole patterns and other work were done on a CNC mill. IDK if its possible to do it with just a lathe. Outsourcing stuff will add cost quickly.
- Back to the group thing. This took around 10 senior engineers a year to develop, lead by one of the best students our school has seen in a while. Now our group is doing the programming of the servo actuated valves for flight, which is being heavily assisted by a few electrical engineers. Again, an insane amount of knowledge/expertise.
- I would stay away from LOX/cryogenics. Despite all this, the engine above does not run on cryogenic fuel/oxidizer. That is a whole 'nother ballgame.
I'm not trying to scare you, just caution you about what you are getting into. Good luck with the endeavor.