For the 223 for deer, if you HAVE to, use as heavily constructed bullet that is accurate.
That said, try bullets from 40gr on up to 55. There is one Sierra 63 gr that has produced acceptable accuracy, but mostly 55 and under.
For powder, start with BL-C(2) and H335, as these seems to produce the best overal results in the Handi.
My particular best is 27gr of H335 with Hornady 40 Vmax. For factory stuff the Win White box (40 pak) seems about the best. If the rifle is new, make sure you clean the ejector assembly and barrel and chamber very well BEFORE shooting. You might also consider polishing the chamber and bore too. I did on mine and breakin time was drastically reduced. Plus have never had a stuck case yet.
Otherwise it may take 200-300 rounds before it settles down to good groups.