Should i see a doctor due to cutting a vein on my pinky finger and now my finger is turning purple?

Yes: Depending on the location and the depth of the cut, you could have injured a vein, or deeper structures such as artery or nerve. Based on your descroption, i doubt that you could have injured tendons, bone or joint but only a physical examination can determine this.

Finger injury: It is difficult to determine the extent of injury without an examination by a physician. You may have only caused some bruising if it was a superficial cut. If deep, however, it is best to have a doctor look at it soon.