Counting the position of ones in a binary vector

only as guide, not tested

process(data_in)
begin
for i in 0 to 15 loop
    if data_in(i) = '0' then
          data_out(i*4:i*4+3) <= "0000";
     else
          data_out(i*4:i*4+3) <= std_logic_vector(to_unsigned(i,4));
     end if;
end loop;
end process;

This will not be straightforward to realise in a combinatorial circuit because what you describe is a rather sequential process: traverse the input vector from right to left, if you encounter a bit set to '1' add its position index into the next available output slot.