In my opinion the best reference is by far Grothendieck's La théorie des classes de Chern.
This seminal article was published in 1958, is purely algebraic and is valid in characteristic $p$.
Needless to say it doesn't necessitate any differential geometry: no curvature of connections here!
This article was written before Grothendieck introduced scheme theory and is incredibly elementary, probably the simplest text he has ever written!
It relies on the purely geometric idea that given a vector bundle $E$ on the variety $X$, you should consider the associated projective bundle $\mathbb P(E)$ over $X$, lift $E$ to $\mathbb P(E)$, quotient out the tautological line bundle and iterate.
This idea is "childish", an adjective Grothendieck loves to apply to his work, and incredibly powerful.
Even differential geometers/algebraic topologists use it: Bott and Tu introduce characteristic classes by means of Grothendieck's construction in their celebrated Differential Forms in Algebraic Topology.
And incidentally their treatment is also an excellent introduction to Chern classes: the ideas are from Grothendieck but there are more applications/examples in their book.