hex_table db '0123456789ABCDEF'
...
mov esi,source
mov edi,destination
mov ecx,source_length
cld
@@:
lodsb
mov dl,al
and eax,0x0F
mov al,[eax+hex_table]
stosb
mov al,dl
shr al,4
and eax,0x0F
mov al,[eax+hex_table]
stosb
loop @b    

hex_table:
repeat 256
 l = (%-1) and 0Fh
 h = (%-1) shr 4
 if h < 10
  db '0'+h
 else
  db 'A'+h-10
 end if
 if l < 10
  db '0'+l
 else
  db 'A'+l-10
 end if
end repeat

...

mov esi,source
mov edi,destination
mov ecx,source_length
cld
xor edx,edx
@@:
mov dl,[esi]
inc esi
mov ax,[hex_table+edx*2]
stosw
loop @b