Unless you're an aeronautical engineer, I would not suggest this. If you want to build a jet, there are plans available. This gets you past the first hurdle, proving the design WILL fly. Start by googling for the BD5J, the jet version of the BD5.
Once the FAA approves the design, you can start building.
If you mean "ultralight" as in a cloth-wing tubular frame aircraft - well, the problem with jet engines is that they're not terribly efficient until you hit 200+mph speeds.
I would also suggest that you do what most "personal jet" builders do, and use an existing turbine unit. The ones in BD5 jets and similar are from ground-based powerplants or discards from drones, but put out enough thrust (about 80lb, IIRC?) for a one-man jet. They probably cost about $10K to $50K...
Building your own turbine is not a good idea for a project until you can build V8 auto engines from a block of metal in your sleep. Turbines are high-temperature high-speed spinning precision pieces of equipment. The metal castings have to be flawless, the metal used is special to tolerate the high temperatures and to minimize expansion during high heat and the blades are machined to complex curves.
Even the first one built by engineers in the 1940s had the habit of flying apart or bursting into flames. Sometimes the "don't try this at home kids" thing really means what it says.
If you mean a "pulse-jet" engine, like the old V1 German flying bombs from WWII - the "build your own jet engine" stuff you see in the back of Popular mechanics - that is also not very efficient. For an experimental aircraft running off normal airports, as opposed to an ultralight or sport plane, you need to have 45 minutes fuel reserve when you land. Pulse jets usually run off propane and a tank will not last even that long.