On which speed the relic microwave background radiation becomes dangerous for the people on space ship?

If we travel with speed close to the speed of light, the light waves becomes shorter and their energy increases (because we flight through much more light for the same time).

So, even background radiation can turn into x-ray and gamma on some speed. So, what is the maximum speed with which we still can travel safely without a lot of radiation produced by relic microwave radiation?

You can look up the peak frequency of the cosmic microwave background and the frequency where X-rays begin to become dangerous. Then you can use the formula for the relativistic Doppler effect $$\nu_\text{observed} = \nu_\text{source} \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$$ where $$v$$ is the velocity between observer and source, $$c$$ is the speed of light, $$\nu_\text{observed}$$ is the observed X-ray frequency, and $$\nu_\text{source}$$ is the cosmic microwave frequency.
Solve this equation for $$v$$, plug in the numbers, and you will get a velocity very close to speed the of light.