What is the relationship between thrust and engine RPM in a turbojet?
First, lets see how thrust works. NASA explains it pretty good. They conclude that for turbo-fan engines "specific thrust [...] depends only on the velocity difference produced by the engine." If you think about it, velocity difference in a gas is dynamic pressure. So, how much thrust the engine will generate depends on how big of a dynamic pressure difference it can create in the air between its front and back. A turbo-fan engine creates this pressure difference in two ways. One is by using a fan to "push" air to the back. The other one is by heating up the air that passes through its burner and shooting it out of the nozzle (after passing it first from the turbine.)

Now, on to your question. We need to see if added RPM creates added thrust (or if more thrust needs more RPM.) The answer is that, yes it does. Providing the engine with more fuel will create faster escaping hot gasses from the burner. Those will create a bigger dynamic pressure difference between the burner and the nozzle and the turbine will turn faster. Power from the turbine is moved to the fan which will also turn faster thus pushing out more air.

Refer to here for a graphical description of the engine compartments and how pressure is formed between them.

How RPM and total thrust for a model turbo-fan engine are related is graphed on page four and seven of this document. There you can also find various equations that describe those graphs.

EDIT: A turbo-jet engine works pretty much the same but instead of using the power generated from the turbine to mainly move the fan, it uses it to move only the compressor. All its thrust is generated from the hot gasses coming out of the nozzle.