Bic Ink and other ink in ink pens and markers is considered to be slightly toxic especially if you were to ingest the ink and then it could harm or kill you if you consumed enough of it.
Or if the ink from Bic pens or markers got into your bloodstream through your skin in large enough amounts then it could lead to ink poisoning through your blood.
But you would have to saturate your skin with about a gallon or so of the Bic or marker ink for it to cause ink poisoning through your blood.
If you just write on yourself using markers such as Sharpie or Bic ink pens or other ink pens there's no harm in getting ink poisoning from writing on your skin.
I've wrote on my skin for years with Sharpie and Bic Ink pens as well as many other ink pens and it never caused me any health issues.