This is a standard situation for such kind of theorems, because the proof of existence is not a construction, but an application of Zorn's lemma. Even for simpler objects, like the algebra of all bounded sequences, there is no any explicit formula for complex homomorphisms  (whose kernels are maximal ideals). There are descriptions, that look like explicit formulas  (like limit with respect to an ultrafilter, or integral with respect to a finitely additive measure defined on all subsets of naturals, and taking only values 0 and 1), but the corresponding objects, like free ultrafilter, are not constructive in principle. It is not just a fault of the known techniques of the proof, but an essential property of objects like ultrafilters. Say, one can prove that if one considers the topological space 2N of all subsets of naturals, then every free ultrafilter on N is a non-Borel subset of 2N, which means that there are no any ``usual'' formula, that describes a free ultrafilter.

Thank you. I was afraid that the answer is no, but you explained the reason for that. On the other hand there is a group of homomorphisms indexed by real numbers t: the value at t of an almost periodic function. I hoped some other homomorphisms are known; maybe related to Fourier spectrum. After your answer, I suppose all other homomorphisms are hidden in the way you present (Axiom of Choice) and Bohr compactification is, roughly speaking, uniformly distributed along the real line, completely opposite to the case of Alexandroff's compactification.

You did not precise if you consider the algebra of CONTINUOUS almost periodic functions, as it is supposed usually. If so, I would suspect that a typical complex homomorphism, that is not an evaluation functional, reflects behavior of elements at infinity. Say, if one takes an arbitrary tending to infinity sequence tn  of reals, and an arbitrary ultrafilter U on N, then the functional F that maps every bounded almost periodic function f to the U-limit of f(tn) is a complex homomorphism. If the algebra that you consider contains also discontinuous functions, there will be also homomorphisms similar to "limit at a point": just take in the previous  example tn tending to some finite point.

I don't have an answer to your question. Since you seen interested in almost periodic functions, let me call your attention to how these functions characterize certain ideals in the Banach algebra L1(R) (under convolution). As you know, spectral synthesis fails in this algebra so, given a closed ideal J we say that J can be synthesized if it is the intersection of the regular maximal ideals that contain it. Then: J can be synthesized if, and only if, there is a (Bohr) almost periodic function b  such that 

J = { f in L1(R) : f convolved with b is equal to zero}.  The details can be found in the paper "Chacterizations of those ideals in L1(R) which can be synthesized"  Math. Ann. 203, 171-173 (1973).