With his grandparents in 1871, His parents in 1881. Name is a little to common to trace with certainty after this point.
Sources
Birth: gro.gov.uk, GRO Reference: 1860 D Quarter in ASTON Volume 06D Page 255
1871 Census: "England and Wales Census, 1871", database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:VRFQ-3X5 : 10 December 2017), William Jenkins in entry for John Blackwell, 1871.
1881 Census: "England and Wales Census, 1881," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:Q27N-7SN7 : 13 December 2017), William Jenkins in household of David Jenkins, Birmingham, Warwickshire, England; from "1881 England, Scotland and Wales Census," database and images, findmypast (http://www.findmypast.com : n.d.); citing p. 37, Piece/Folio 2978/22, The National Archives, Kew, Surrey; FHL microfilm 101,774,842.
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DNA Connections
It may be possible to confirm family relationships with William by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with William: