What is the source of the fat in adipose tissue?
The proximal source of adipocyte lipids is mainly fatty acids from circulating lipoproteins (1) after hydrolysis by lipoprotein lipase (LPL). 

LPL is activated by ApoC-II, which is present in hepatic-originating VLDL and IDL lipoprotein, but also chylomicron of direct dietary source. Insulin, secreted after meals, stimulates LPL production by adipocytes.

Thus the ultimate source of these lipids is both hepatic lipogenesis (from other substrates like glucids) and dietary fats.

Lipid transport in blood, for reference purpose (Michal G. Schomburg D., Biochemical Pathways, 2012)