I am addressing only the first part of your question (i.e., nothing with the structure of vector space; only topology and limits of sequences).
I will quote here part of Problems 1.7.18-1.7.20 from Engelking's General Topology. (It would be better if you could get the book. I believe it used to be here, but the links don't work now. Perhaps you'll find it in the Internet.)
L*-space is a pair $(X, \lambda)$, where X is
a set and $\lambda$ a function (called the limit operator) assigning to some sequences of points of X
an element of X (called the limit of the sequence) in such a way that the following conditions are satisfied:
(L1) If $x_i=x$ for $i = 1,2,\dots$, then $\lambda x_i = x$.
(L2) If $\lambda x_i = x$, then $\lambda x' = x$ for every subsequence $x'$ of $x$.
(L3) If a sequence $\{x_i\}$ does not converge to $x$, then it contains a subsequence $\{x_{k_i}\}$ such
that no subsequence of $\{x_{k_i}\}$ converges to $x$.
These properties are sufficient to define a closure operator on $X$ (not necessary idempotent).
If $(X,\lambda)$ fulfills and additional condition
(L4) If $\lambda x_i = x$ and $\lambda x^i_j = x_i$ for $i = 1,2,\dots$, then there exist sequences of positive integers
$i_1, i_2,\dots$ and $j_1, j_2, \dots$ such that $\lambda x_{j_k}^{i_k} = x$.
L*-space $X$ satisfying (L4) is called an S*-space. The closure operator given by S*-space is idempotent.
Using this closure operator we get a topology, such that the convergence of the sequences is given by $\lambda$. A topology can be obtained from a L*-space (S*-space) if and only if the original space is sequential (Frechet-Urysohn).
References given in Engelking's book are Frechet [1906] and [1918], Urysohn [1926a], Kisynski [i960].
Frechet [1906] Sur quelques points du calcul fonctionnel, Rend, del Circ. Mat. di Palermo 22 (1906), 1-74.
Frechet [1918] Sur la notion de voisinage dans les ensembles abstraits, Bull. Sci. Math. 42 (1918), 138-156.
Kisynski [1960] Convergence du type L, Coll. Math. 7 (1960), 205-211.
Urysohn [1926a] Sur les classes (L) de M. Frechet, Enseign. Math. 25 (1926), 77-83.
NOTE: Some axioms for convergence of sequences are studied in the paper:
Mikusinski, P., Axiomatic theory of convergence (Polish), Uniw. Slaski w Katowicach Prace Nauk.-Prace Mat. No. 12 (1982), 13-21. I do not have the original paper, only a paper which cites this one; it seems that the axioms are equivalent to (L1)-(L3) and the uniqueness of limit. (But I do not know, whether some further axioms are studied in this paper.)
EDIT: In Engelking's book (and frequently in general topology) the term Frechet space is used in this sense, not this one. I've edited Frechet to Frechet-Urysohn above, to avoid the confusion.
The definition given by Munkres is correct. The set $B_C(f,\epsilon)$ contains the functions $g:X\to Y$ for which $\sup_{x\in C}d\big(f(x),g(x)\big)$ exists and is less than $\epsilon$. If $g:X\to Y$ is such that the supremum doesn’t exist (or if you prefer, is infinite), then $g\notin B_C(f,\epsilon)$, that’s all.
The topology of compact covergence is defined in Wikipedia; the definition is given in terms of which sequences of functions converge rather than directly in terms of the topology, but if you compare it with Theorem $46.2$ in Munkres, you’ll see that it’s the same topology.
Both the uniform topology and the topology of compact convergence are extremely useful and widely used.
Best Answer
In general the definition of a notion of convergence for sequences will not be enough to define all topologies, but only topologies that can be described by sequences (so-called sequential topologies). There will be some conditions that the sequence convergence definition/criterion has to meet, the most trivial one being that a constant sequence has to converge to that constant value.
In your case, I think most conditions will be met and we can define a set $C$ to be closed under the weak-star topology iff for all sequences where both $x_n \to x$ (under this definition) and and all $x_n \in C$, then $x \in C$ too.
Then this will probably obey the axioms for closed sets of a topology, under some conditions on $X$ or $C(X)$. Normally though, the open sets in weak-star topology are defined directly without convergence.