Evensson was born in Torsby and his family moved from Torsby to Hagfors when Conny was just two years old. He started his hockey career with Hagfors club IK Viking before signing with Malmö FF in 1965.
Evensson was born in Torsby and moved to Hagfors at a young age, beginning his hockey career with the local club IK Viking before joining Malmö FF in 1965.