The vivisectionist's cruel anatomist ability allows him to "use his Knowledge (nature) skill bonus in place of his Heal skill bonus". Does this mean that using this ability while making a check ignores the +2 circumstance bonus from a healer's kit? If so, would the kit would still be required if one wanted to avoid the -4 penalty to treat deadly wounds?
The healer's kit works as normal in this case.
The key to this is the wording of the Cruel Anatomist ability (emphasis mine):
At 3rd level, a vivisectionist may use his Knowledge (nature) skill bonus in place of his Heal skill bonus.
Note what it says: you don't make a Nature check instead of Heal check, you use your Nature skill bonus for a heal check. You may be using your Nature skill bonus, but the check you are making is still a Heal check.
Now let's look at Healer's Kits (emphasis mine):
This collection of bandages and herbs provides a +2 circumstance bonus on Heal checks. A healer’s kit is exhausted after 10 uses.
A healer's kit gives its bonus when used for a Heal check. It doesn't care what skill bonus you actually used, only that the check was intended for the Heal skill.
Treating deadly wounds is a Heal check, and requires the use of a healer's kit. The fact that you used your Nature skill bonus instead of your Heal skill bonus does not modify that.
What is a bonus, anyway?
Part of the issue here is that the use of the word "bonus" in the Cruel Anatomist ability is vague, so it's difficult to say exactly what is and is not included. In general, bonuses will either apply to skills or to skill checks. In this case, Cruel Anatomist lets you substitute the Nature skill for the Heal skill, but you're still making a Heal check. Effects that directly increase your Nature skill (Int mod, skill ranks, class skill bonus) will apply when treating deadly wounds with Cruel Anatomist, as will effects that apply a bonus to Heal checks (such as a healer's kit), while effects that apply a bonus to Nature checks will not apply (most racial bonuses apply to checks).
A murkier issue would be things like Skill Focus, which gives a bonus on "checks involving the chosen skill", or most skill-boosting backgrounds, which give bonuses on "checks using that skill". Thankfully, resolving those issues is not relevant to the question at hand.