Based on Dr.’s directives I take liver pills, which is basically a capsule that contains a kosher drive liver inside. My question is whether I am considered Fleishig after swallowing the pill, and may not drink milk for six hours after?

You are not considered Fleishig, and may eat dairy afterwards, even right away. However, do not use milk to swallow down the pill.

Sources: Igros Moshe Y.D. 2:26; Rivivos Efraim 5:517; Beir Moshe; Pesakim Uteshuvos 89:7 footnote 74; Yakut Yosef 89:28