How could I determine if I have dislocated a joint?

X-ray. Best way would be to have an x-ray taken.
Dislocations. Dislocation is usually accompanied by severe pain and distortion of the normal anatomy and loss of function. If you believe you have a dislocation of any joint then you should consult with your doctor to be examined and then advised to the best treatment.
X- ray. X- rays are definitive. Another condition, called subluxation, is more difficult. Subluxation is a partial "dis-location" that shifts in & out of normal anatomical alignment. For example a large joint (sacroiliac) subluxes & medical literature reports no imaging study able to show. Reason: subluxation is a motion & all imaging studies are "snap shots". Cinemagraphic fluoroscopy failed also.