By following the plater Gustaf Hellström (b. 7 April 1847 in Stockholm) backwards, from his son Josef's birth 1879 in Karlstad, I have found him listed as a son of Sara Andersdotter Hellström b. 24 September 1805 in Karlskoga, a widow since 30 October 1855. Gustaf had a brother Erik b. 11 April 1849 in Finland. In 1860 Sara Andersdotter Hellström and her son Gustaf moved to Sund, Nysund parish, Värmland county, and Erik Hellström moved to Kil parish, also Värmland county.
(Source: Household examination of Rudskoga parish, Värmland county, 1856-1860 page 284, place S. Gren)
Unfortunately, I could not find Sara and her sons at S. Gren in the previous hh ex of Rudskoga (1851-1855).
Gustaf Hellström's way from Nysund to Karlstad: (Some of these records are available online only at Genline, some other only at SVAR)
Nysund hh ex 1861-1865 p. 360 (Sund)
Nysund hh ex 1866-1870 p. 149 (Forsnäs), p. 232 (Roted), p. 235 (Rotedsbäcken)
Nysund hh ex 1871-1875 p. 236 (Rotedsbäcken)
Lundby, Göteborg's and Bohus' county, hh ex 1869-1876 p. 293.
Maria Magdalena parish, Stockholm city, migration roll 1875, letter H nr 31: Gustaf Hellström moved in 26 June 1875 from Lundby
Karlstad stadsförsamling (town parish), Värmland county, hh ex 1871-1875 p. 37
Karlstad stadsförsamling hh ex 1876-1880 p. 771
(Meddelandet ändrat av mausve den 13 september, 2008)