The Delton will do just fine. The ammunition will be key. Top pick for deer are budsgunshop.com/catalog/product_info.php/products_id/85839/Fiocchi+Rifle+Ammo+Extremma+223+Rem%2F5.56+NATO+V-Max+50GR+200 & budsgunshop.com/catalog/product_info.php/cPath/656_2081_10000601/products_id/59189/Federal%20F223FS1%20Fusion%2020RD%2062gr%20223%20Remington/ & budsgunshop.com/catalog/product_info.php/cPath/656_2081_10000601/products_id/712012548/BAR%205.56%2062GR%20TSX%20BT%20VOR-TX%2020/10/ & budsgunshop.com/catalog/product_info.php/cPath/656_2081_10000601/products_id/11122/Federal%20P223Q%20Vital-Shok%20Nosler%20Partition%2020RD%2060gr%20223%20Remington/

The biggest problem when using the .223/5.56 for big game is bullet placement and performance. My choice for .223 ammo is Barnes. You must first determine if it is legal to use your .223 for deer hunting then learn to kill the animal. Study diagrams and get targets that show the vital organs of the deer and practice putting your bullet in the lungs or heart to kill them. The .223 does not have much power so you must learn to precisely place your bullet for a quick kill.
budsgunshop.com/catalog/product_info.php/products_id/69471/Barnes+VOR-TX+223+Remington%2F5.56+Nato+TSX+Flat+Base+55+GR+20

while the .223 and 5.56 will kill deer they are far from a ideal caliber for that task. They are also illegal for use on deer in many states so check your state/local game laws 1st

Here is a great article that covers it http://blog.del-ton.com/2012/05/feeding-your-ar-15.html

DTI AR-15 rifles are chambered in 5.56X45 NATO, so both 5.56 and .223 Remington ammunition is safe to use in them. 5.56 ammunition typically has more velocity and higher chamber pressures than .223 ammunition. This potentially higher chamber pressure and size differences in the leade (throat) of the two weapons are reasons it is dangerous to use 5.56 ammunition in a weapon chambered only for .223 Remington. The selection of 5.56 ammo is a bit more limited, though recent military activities and developments have increased available options. Not long ago, almost all 5.56 ammunition was loaded with "ball" ammunition/FMJ (full metal jacket). There are now a few different manufacturers that produce 5.56 ammo with "open tip match" or expanding bullets.
Ammunition in .223 is also fine to use in your AR-15 and the available ammunition options are much greater in this caliber. From FMJ to expanding solids to various hollow point designs, a perfect option for just about any shooting application can be found.
Both 5.56 and .223 ammunition is available in a variety of bullet weights (grains). Typically, the lighter grain bullets are more suited for varmint hunting and target shooting/plinking, while a heavier grain bullet will perform better in situations like deer hunting or self defense. There is a line of thought that the lighter weight bullets are more accurate with slower twist rate barrels (1X9), while heavier bullets perform better in a faster twist rate (1X7).