Claudio Baca was born in February of 1851, according to information on the 1900 US census for San Rafael, Valencia County, New Mexico Territory.
Claudio can be found on the 1900 US census for San Rafael, Valencia County, New Mexico Territory, US, at the age of 49, along with his wife Sara (Landavazo) Baca and their children. According to the census, they had been married for 11 years at the time. The census also states that she was the mother of 7 children, five of whom were alive when the census was taken. Every person in this household is listed as having been born in New Mexico, and each individual's mother and father are shown as having been born in New Mexico.
Claudio Baca, Male, age 49, Married, Head, b. Feb 1851 in New Mexico
Sara Baca, Female, age 31, Married, Wife, b. Jul 1869 in New Mexico
Sirila Baca, Female, age 10, Single, Daughter, b. Jun 1890 in New Mexico
Juan Baca, Male, age 9, Single, Son, b. Jun 1891 in New Mexico
Traquilino Baca, Male, age 7, Single, Son, b. May 1893 in New Mexico
Terecita Baca, Female, age 3, Single, Daughter, b. Apr 1897 in New Mexico
Elisea Baca, Female, age 1, Single, Daughter, b. Mar 1899 New Mexico
Claudio is likely (but not certain to be) the person by that name on the 1860 US census for Valencia Territory, New Mexico Territory.
Juan B Baca Male 31 White 1829 New Mexico 213
Teresa Baca Female 25 White 1835 New Mexico 213
Ludovino Baca Male 10 White 1850 New Mexico 213
Josefa Baca Female 9 White 1851 New Mexico 213
Claudio Baca Male 5 White 1855 New Mexico 213
Cecila Baca Female 4 White 1856 New Mexico 213
Macedonio(?) Baca Male 2 White 1858 New Mexico 213
If this is the correct Claudio, his parents are likely to be Juan B. Baca and Teresa Baca. 
The same family can again be located again on the 1870 US census (minus Juan):
Teresa Baca Female 38 White 1831-1832 New Mexico 10
Josefa Baca Female 16 White 1853-1854 New Mexico 10
Claudio Baca Male 15 White 1854-1855 New Mexico 10
Macedonio Baca Male 12 White 1857-1858 New Mexico 10
Braulia Baca Female 10 White 1859-1860 New Mexico 10
Domingo Baca Male 8 White 1861-1862 New Mexico 10 
Claudio can be found again with his wife, children and a grandchild on the 1920 US census for Magdalena, Socorro County, New Mexico:
Claudio Baca Male 63 Married White Yes Yes Head Own 1857 New Mexico New Mexico New Mexico
Sara Baca Female 51 Married White Yes Yes Wife 1869 New Mexico New Mexico New Mexico
Elisea Baca Female 19 Single White Yes Yes Daughter 1901 New Mexico New Mexico New Mexico
Leopoldo Baca Male 17 Single White Yes Yes Son 1903 New Mexico New Mexico New Mexico
Crusita Baca Female 11 Single White No No Daughter 1909 New Mexico New Mexico New Mexico
Brulio Baca Male 9 Single White No No Son 1911 New Mexico New Mexico New Mexico
Claudio Sanchez Male 11 Single White No No Grandson 1909 New Mexico New Mexico New Mexico
It may be possible to confirm family relationships with Claudio by comparing test results with other carriers of his Y-chromosome or his mother's 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 DNA with Claudio: