You will probably be around 2 grains above any published load data you find seeing as that the SAAMI pressure limits are lower than the Nato pressure limits.
H335 will work (very flashy and temperature sensitive) although you may want to try the CFE223 powder as it seems to advertise higher velocities for the same pressure. AA2230 is supposed to be 'the' powder for loading 5.56 but it seems hard/expensive to get at the best of times.
Make sure you have a 5.56 chamber (throat). Use a chronograph when working up the load. Use CCI 450 or CCI 41 primers. Keep an eye on those primers, if you go to high you will be pierce them or be picking them out of your fire control group and maybe your teeth (remember, your primers won't be crimped in place).
Also make sure your bullets are crimped. You don't want the possibility of a set-back bullet ruing your day and rifle.
As an aside, I have found that PPU 55gr ammo seems to match the velocity you would expect to see in M193 rounds (I know, not SS109 or M855) and it does seem like a way hotter load than your typical AE ammo. I don't know if they make a 62gr variety.