you wanted count to go from 0 to 255 and back, yet you are asking if count > 255 which will never occur in your new code.
Here is code for my understanding of your plans earlier in your post
entity counter is
port (clk : in std_logic;
on_input : in std_logic;
reset : in std_logic;
overflow : out std_logic;
values : out integer range 0 to 255);
end counter;
architecture Behavioral of counter is
signal count : integer range 0 to 255;
signal on_input_d: std_logic := '0';
begin
process (clk, reset)
begin
if reset = '1' then
count <= 0;
on_input_d <= '0';
elsif rising_edge (clk) then
on_input_d <= on_input;
if on_input = '1' and on_input_d = '0' then
count <= count + 1;
end if;
end if;
end process;
process (count)
begin
if count = 255 then
overflow <= '1';
else
overflow <= '0';
end if;
end process;
values <= count;
end Behavioral;