This is a bit tangential to your query but I hope that it might be useful. The duality for Banach spaces that you mention is, in my opinion, best expressed in terms of the symmetric one between between Banach spaces and Waelbroeck spaces. The latter concept is due to Waelbroeck and Buchwalter and is perhaps easiest accessible in the classic text by Cigler, Losert and Michor ("Functors on Categories of Banach Spaces"). A convenient way to look at it is in the context of the pro and ind categories (Grothendick et al.). The category of Banach spaces (with linear contractions as morphisms) is the ind completion of the finite dimensional ones, that of Waelbrock spaces the pro-completion so the duality follows by abstract nonsense from the finite dimensional case.
With regard to the spaces of continuous functions, firstly I am of the opinion that the natural framework is that of (functionally separated) compactological spaces (again due to Buchwaltere, based on work by Waelbroeck) and the so-called Saks spaces. The latter are Banach spaces with a suitable auxiliary l.c. topology on the unit ball. One then considers a duality between a compactological spaces and the bounded continuous functions thereon, regarded as a Saks space with the uniform norm and the topology of compact convergence. The details are too convoluted to expose here but can be found in the monograph "Saks spaces and Applications to Functional Analysis (first edition)". Once again, this duality is natural when considered in the context of pro and ind categories (applied to the categories of compact resp. Banach spaces).
One can also presumable develop a suitable duality in terms of (not necessarily) bounded continuous functions as in your query, it is just that nobody has, to my knowledge, actually sat down and done this.