Does Hilbert's Nullstellensatz have something to do with the universal property of polynomial rings? One basic slogan of category theory is that a mathematical concept is uniquely defined by a universal property. For example, polynomial rings are uniquely characterized by their universal property.
Does this mean that every statement about polynomials can be deduced from this universal property? For example, consider the following statement about polynomials, the so-called Hilbert's Nullstellensatz (formulation copied from The Princeton Companion to Mathematics):

Nullstellensatz. The polynomials $f_1,...,f_m$ have no common complex zero if and only if there are polynomials $g_1,...,g_m$ such that $g_1f_1 +···+ g_mf_m = 1$.

Does this statement have something to do with the universal property of polynomial rings? Or should one distinguish between properties of polynomial rings (these should be derivable from the universal property) and concrete statements about polynomials such as Hilbert's Nullstellensatz?
 A: The problem, as KReiser says in the comments, is that the universal property of polynomial rings requires no hypotheses on the base $k$, whereas in the Nullstellensatz $k$ is traditionally an algebraically closed field (in the form you've stated it) and more generally can be a Jacobson ring (in a different form which turns out to be equivalent when specialized to an algebraically closed field). The proof must make use of this extra hypothesis somehow, which is extra "non-categorical input" specific to the situation.

One basic slogan of category theory is that a mathematical concept is uniquely defined by a universal property.

As much as I love category theory, this is much too strong of a slogan. Many interesting mathematical concepts have nothing to do with universal properties.
Generally speaking, at the most basic level, knowing the universal property of an object tells you what either maps into or out of that object look like, depending on the universal property. In principle this completely determines the object but it may not be the best way to prove facts about it, depending on how easily those facts can be stated in terms of maps in or out. Again at the most basic level, if you know a maps-in universal property then that doesn't necessarily help you figure out what maps out look like, and vice versa.
Let me switch to the simpler example of groups so I can be explicit here. The free groups $F_n$ have a maps-out universal property, which is useful for proving various facts involving maps out. For example the maps-out universal property directly implies that $\text{Hom}(F_n, \mathbb{Q}) \cong \mathbb{Q}^n$, which implies that the free groups are not isomorphic for different values of $n$. It also implies that the abelianization is $\mathbb{Z}^n$ and so forth.
Consider, by contrast, statements like "$F_n$ is torsion-free" or "every subgroup of $F_n$ is free." These are statements about maps in, not maps out, and the universal property is less helpful here. In fact a "fully universal" proof is impossible because analogues of these statements are false in other categories of algebraic structures (e.g. subrings of polynomial rings are not polynomial). So at some point one must make use of some "non-categorical input" specific to this situation.
The Nullstellensatz itself is equivalent to the statement that a polynomial ring over a field (or more generally a Jacobson ring) is Jacobson, meaning that every prime ideal is an intersection of maximal ideals. This is in principle a maps-out statement, although a somewhat complicated one, and the polynomial ring has a maps-out universal property, but I'm not aware of a way of proving it in which the universal property really helps at all.
