Not sure what kit you are looking at. Can you post a link?
Regardless, to make the jump to all-grain you will need (assuming you don't already have these items):
1. Brew kettle that is large enough. Assuming 5.5 gallon batches, I'd recommend no smaller than 10g kettle. 12g would be better. You can get away with 8g if doing 60 minute boils, but it will be tight for 90 min or longer boils.
2. Wort chiller
3. Accurate thermometer for mashing
4. Mash/Lauter Tun (converted cooler or kettle w/false bottom) for traditional all-grain or a brew bag if doing BIAB. If you're just getting into it for the first time, I'd recommend BIAB, but it's up to you. If you do BIAB, invest in a proper bag. Forum user Wilserbrewer makes and sells quality bags for a reasonable price.
5. Ferm temp control and if using liquid yeast, the ability to make yeast starters. Not necessarily required for extract or all-grain, but your beer will be much better if you have these. I'd make sure you have this covered before investing in all-grain equipment.
6. Grain mill. You can get away with not having your own if you have a LHBS nearby that can crush your grain or if you order it online pre-crushed, but ultimately, you'll want to have control over your crush to minimize efficiency inconsistencies due to variations in crush from batch to batch.