They believe in Oedipus, he is their king.
There are three ways to solve this problem:
They believe in Oedipus. He is their king.
They believe in Oedipus, for he is their king.
They believe in Oedipus; he is their king.
In this particular example, using the first option would result in sentences that are too short, too choppy. There is an obvious connection between the clauses which is best expressed through a conjunction; therefore, the second option is the best to use here.
In the following example, however, the writer has used a comma where a semi-colon is appropriate:
Sgnarelle is not the primary character, still he acts as a foil.
The clauses could form separate sentences, but the use of "still" implies a stronger link which is best served by a semi-colon:
Sgnarelle is not the primary character; still, he acts as a foil.
In this example, the clauses being connected are too complex to be part of the same sentence:
Lear was a majestic ruler when he was young, however, as he became older, his temptations clouded his thought.
Lear was a majestic ruler when he was young. However, as he became older, his temptations clouded his thought.
The writer could use a conjunction:
Lear was a majestic ruler when he was young, but as he became older his temptations clouded his thought.
A comma cannot, by itself, connect two main clauses; the clauses must either form separate sentences or be joined by a coordinating conjunction. For main clauses, see the Short Treatise on Grammar and the "tip" after "sentence fragments" in this section.
Note that conjunctive adverbs have the logical effect of linking ideas, but do not link sentences grammatically. In the last example the word however is one of these. Others to watch for are therefore, thus, and this.