The analysis of a cluster environment is a valuable instrument to investigate the origin of gas fuelling and trigger mechanisms in active galactic nuclei (AGN) and star-forming galaxies. To this purpose, we present a detailed analysis of the point-like X-ray sources in the Bullet cluster field. Based on similar to 600 ks Chandra observations, we produced a catalogue of 381 X-ray point sources up to a distance of similar to 1.5 virial radius and with flux limits similar to 1x10(-16) and similar to 8x10(-16) erg cm(-2) s(-1) in the 0.5-2 keV and 2-10 keV bands, respectively. We find a strong (up to a factor 1.5-2) and significant (>= 4 sigma) over-density in the full region studied 0.3R(200)< R< 1.5R(200). We identified optical (R band) and infrared (Spitzer IRAC) counterparts for similar to 84% and similar to 48% of the X-ray sources, respectively. We obtained new spectroscopic redshifts for 106 X-ray sources and collected from the literature additional 13 spectroscopic and 8 photometric redshifts of X-ray sources. Twenty-nine X-ray sources turned out to be cluster members. Spectroscopic and photometric redshifts of optical and infrared sources were also collected, and these sources were used as ancillary samples. We used these multi-wavelength data to characterise the nature of the Bullet cluster X-ray point sources. We find that the over-density in the region 0.3R(200)< R< R-200 is likely due to X-ray AGN (mostly obscured) and star-forming galaxies associated with the cluster, while in the more external region this over-density is likely to be mostly due to background AGN. The fraction of cluster galaxies hosting an X-ray detected AGN is 1.0 +/- 0.4%, which is nearly constant with the radius; this fraction is similar to that reported in other clusters of galaxies at similar redshift. The fraction of X-ray bright AGN (L2-10 keV> 10(43) ergs s(-1)) in the region 0.3R(200)< R< R-200 is 0.5(-0.2)(+0.6)%0.5-0.2+0.6%, which is higher than that in other clusters at similar redshift and more similar to the AGN fraction in the field. Finally, the spatial distributions of AGN and star-forming galaxies, which we selected for their infrared emission, appear similar, thus suggesting that both are triggered by the same mechanism.