Since the question involves only classical ideas, it's enough to refer to work of Serre and others explained by Hartshorne in section III.5 of his text on algebraic geometry. In quite a bit of generality, one sees (starting with line bundles on the projective line) that the cohomology groups of a coherent sheaf on a projective variety (or scheme) have finite generation properties. This generalizes the earlier analytic work on manifolds, where the language of vector bundles is used.
EDIT: Maybe it's helpful to quote explicitly the source in Serre's fundamental paper FAC, Faisceaux algebriques coherents, Ann. of Math. 61 (1955), 197-278. (Online access is via JSTOR.) See in particular no. 66, which applies the preceding results on cohomology of coherent sheaves on a projective space to an arbitrary projective variety (relative to an embedding). This way of proving finite dimensionality, together with vanishing above a certain degree, is purely algebraic (and much more general than the case of line bundles) though of course suggested by the parallel ideas for compact manifolds. As Serre commented, his result should conjecturally extend to all complete varieties, but the proof of this required Grothendieck's later work.
While it's fine to work out one's own proof from scratch, it's not clear to me that this will improve on Serre's method.
P.S. This viewpoint was exploited in Invent. Math. papers by Demazure where he revisited Bott's theorem using more algebraic techniques. This shows how to approach the ideas of Borel-Weil and Bott algebraically, which in turn allowed Henning Andersen and others to delve more deeply into what does or doesn't work in prime characteristic for flag varieties. By now there's a lot of literature, some built into Jantzen's monograph Representations of Algebraic Groups (second edition, AMS, 2003). At the outset was Kempf's fundamental vanishing theory for dominant line bundles on flag varieties in any characteristic, which avoided Kodaira vanishing but left a lot of open questions about non-dominant line bundles and other sheaves.