You describe someone's behaviour as social climbing when they try to get accepted into a higher social class by becoming friendly with people who belong to that class.
[disapproval]
All that vulgar social-climbing!
Social climbing is also an adjective.
She was offered the role of the social climbing sister-in-law Debbie.
social-climbing in British English
(ˈsəʊʃəl ˈklaɪmɪŋ)
modifier
relating to advancement to a higher social class, esp by obsequious behaviour
Leroy's ambitious social-climbing wife was talking to Fiona.
social climbing