You use “le/la” before a unique thing. “Je suis la Reine d’Angleterre”
Charlie is possibly unique but it is a name, and you never use “le/la” before a name. As in english, I believe, by the way.

Categories

Tentatively, we should consider five categories:

1. Common names (l’individu, le chien, le chat, etc.)
2. Proper names: personal names (Jean, Jacques), names of places
3. Names of function/title (le duc de Malborough)
4. Antonomasia (la Ville, for Rome or Constantinople, Votre Honneur, Sa Majesté, etc.)
5. Collective names, as members of a community: demonyms (names of people living in a place: les Français) or political group (les Socialistes).

Articles

The rules for articles are rather simple:

1. Common names require an article.
2. In principle, proper names do not require an article. Exceptions are justified by usage: La Rochelle (see no. 4), la Callas (Italian usage for family names, sometimes used in the context of opera).
3. Functions and titles require an article (le président de la République)
4. Antonomasia requires an article or at least a possessive pronoun.
   • La Rochelle, is in fact an antonomasia (its name means “The Rock”)
   • When a personal noun is used as antonomasia, e.g. Michel-Ange for a great painter or sculptor, the rule also applies: Ce n’est pas un Michel-Ange. C’est le Michel-Ange de notre époque. C’est notre Michel-Ange national. If one removes it, however, it becomes emphatic or worse, it could be ironical: Ah, bonjour Einstein!
     5. Members of a group require of course an article (un Socialiste, les Verts).

Capitalization

Note that there are also general rules for capitalization:

1. Common names do not start with capitals.
2. Proper names start with a capital.
3. Functions and titles do not require a capital: la reine d’Angleterre, le duc de Berry (see howerver, no. 4).

4. Antonomasia requires a capital on a noun (and possibly epithet), but not on the article (la Ville) or possessive article (notre Napoléon (à nous)).

   • This may also apply to a title when it refers to a specific person (la Reine, le Tsar, le/notre Président), though this is not a hard and fast rule — it is a matter of deference or intention of the speaker/writer.

   • When a possessive adjective is used in part of an address (Sa Majesté) then it is also capitalized.

     5. For collective names, the capitalization rules are more complicated (often capitalized, but this does not apply to religious groups, e.g. les chrétiens).

Answering your question

So, to answer your question, je suis Charlie:

• It is simply case no. 2, where je suis means je m’identifie à (remember that “Charlie” is short for “Charlie-Hebdo”, a satirical magazine). The rule is no article in that case.

• But if somebody said je suis un Charlie:

  • You could interpret as number 4, as an antonomasia: “I am a person (or magazine?) that is very like Charlie”? There is an illustrious example using Asterix
  • If you used it often enough, then it could even fall into number 5: it would mean that there would be members of a group of people called Les Charlies, whose members took the party of Charlie-Hebdo. There have been instances where this has been used.

This slogan je suis Charlie was used in a very specific context; so it is not a particularly easy thing to understand. If it did take that form, and not the other one (je suis un Charlie) it presumably because it was effective on public opinion in that way. But grammatically, you could have either variants, with slightly different meanings.