It's a massive subject, and there are many different perspectives; here are a few that don't require too much background.
Perspective one: It's a generalization of linear algebra.
Linear algebra is about dealing with systems of linear equations. This is easy: the set of solutions to a (homogeneous) system is just some subspace of $F^n$ (where $F$ is the field of scalars), and you can compute its dimension by row-reducing your system into echelon form.
Algebraic geometry is about dealing with systems of polynomial equations. As you may imagine, this is much harder. In linear algebra, much of the theory is entirely independent of the field $F$, at least until you want to diagonalize operators; in algebraic geometry, non-algebraically-closed $F$ are a massive headache, and there are phenomena in characteristic $p$ that don't show up in characteristic $0$.
Perspective two: It's a computational tool in classical geometry.
In geometry and topology we may wish to study invariants of manifolds. We define lots of invariants, e.g., homology groups, but how can we get our hands on them? For most examples, we can't do it easily at all, but if the example happens to be a complex manifold given by polynomial equations, there's a lot more that we can say. This is especially important if you want to do things with computers.
Perspective three: It's a conceptual way to think about commutative algebra.
If I give you some ring, OK, great, it has prime ideals, maximal ideals, zero divisors, etc. What does all this mean, and how do you ever remember the barrage of technical theorems about integrality, Artin rings, regular local rings, etc?
If the ring is the ring of functions on some space, then the geometry of the space may reflect properties of the ring, and we can remember the commutative algebra by picturing the geometry. What Grothendieck realized is that if we define "space" correctly (which is not so easy), every ring is the ring of functions on some space! For an example of how you might relate geometry to intrinsic properties of the ring: the space attached to a ring is connected if and only if all of the zero divisors in the ring are nilpotent.
No differential geometry or algebraic topology is necessary, though the former might help with motivation for some things, depending upon your mathematical inclinations. You need to know basic graduate abstract algebra (he develops most of the commutative algebra he needs so you don't need to know that in advance) and you need to know some basic point set topology (definition of a topological space, continuous map, Hausdorff, not much more). That's it. For serious.
Would it help to already know some commutative algebra? Yeah, probably, but it's not essential by any means. Liu's is a remarkably self-contained book. I consider it to be an excellent first book on modern algebraic geometry. Whether or not you should first learn some classical algebraic geometry depends on you and your tastes. Liu does do some of that (algebraic sets) in the second chapter. Being familiar with the classical theory might help with motivation, but I don't think it's necessary.
Best Answer
You need some solid commutative algebra. Definitely more than "some of the Commutative Algebra." Without that solid foundation, I think it is just not realistic to "go deep down into the subject." Perhaps not what you want to hear, but some topics are just not accessible with enough background.
I mean, keep in mind that Zariski and Samuel were planning to write a brief intro to the algebra you needed to do algebraic geometry; that ended up being a two-volume book.
The classic intro to commutative algebra at a level suitable to allow you to go into Algebraic Geometry is Atiyah and MacDonald's Introduction to Commutative Algebra, though some people find it too telegraphic. A much more expansive introduction, with examples that would be relevant, is Eisenbud's Commutative Algebra with a view towards Algebraic Geometry.
Both of those presume a solid foundation of abstract algebra, especially rings and modules, as well as some field theory. Neither is for dilettantes.
A further issue is then what algebraic geometry you will want to pursue. The "classic theory" (what is in Hartshorne's first chapter) will not require much more beyond those, but if you venture into the modern theory (sheafs and schemes), you're also going to need some topology... You might be able to work with some small slice of algebraic geometry, say at the level of Fulton's Algebraic Curves, but it's going to be tough going.
When I went to graduate school, Algebraic Geometry was a "second year" subject; it was the rare student who could go straight into it from undergrad. That's one of the issues with Algebraic Geometry: there is a high entrance price.