You usually can prove your kinship to a person through according birth- and marriage certificates. For example your relationship with your brother / sister can be evidenced through the submission of your birth certificate and that of your brother / sister. From this can be seen that both persons have the same parent(s).
If, for example, you or your sister have changed your name due to marriage, you will have to submit your respective marriage certificate.
Please note that the documentary proof of kinship with cousins and other distant relatives is VERY work-intensive.