To answer the last question here, it's certainly not necessary to introduce infinite-dimensional representations into the picture when studying compact Lie groups. On the other hand, I don't quite see the point of trying to relate in isolation the character group of a maximal torus to an abstractly defined "weight" lattice. This is best done within the standard context of finite dimensional representations of a compact Lie group, which of course takes a while to develop from scratch. Look for instance at a standard 1985 textbook by Brocker and tom Dieck Representations of Compact Lie Groups (Springer), where all of the classical theory is laid out systematically.
It's possible of course (as shown by Bourbaki) to treat roots and weights more abstractly without specific reference to Lie groups or Lie algebras, but this is rather artificial here. At any rate, it's risky to start out by identifying the character group of a maximal torus "somehow" with the standard lattice in $\mathbb{R}^n$, since the same could be done with the root lattice or (abstract) weight lattice. It's the placement of one lattice within another that counts.
It's important to have examples at hand, as a reminder that the character group of a maximal torus (say in a semisimple compact group) can vary anywhere between the root lattice (adjoint group) and full weight lattice (simply connected group).
P.S. Added observations: (1) You should assume your group is semisimple (having no central torus of positive dimension) if you want to compare $X(T)$ with the dual lattice of the root lattice, the latter being of finite index in the former. It's also helpful to recall from the general theory that the quotient of these lattices is isomorphic to the center of the simply connected covering group of your given group and to the fundamental group of the adjoint group. (2) As the answers here suggest, your question doesn't have any "easy" answer until you place it in one of the well-established textbook settings for studying structure and representations of compact Lie groups. There are various approaches using analysis, algebraic topology, algebraic geometry, Lie algebras, comparison with complex Lie groups, etc. Your question is mainly pedagogical, so you should first place it in one or more of these traditions.