You can use the phrase "last episode" the same way you'd use "last year," or "yesterday" that is, as an adverb, which means in the most recent prior episode:
Yesterday/Last year, he went to the park -> Last episode, he went to the park
I think he was sick yesterday/last year -> I think he was sick last episode
On yesterday/In last year, he went to the park -> In last episode, he went to the park
I think he was sick on yesterday/in last year -> I think he was sick in last episode
Or, you can use "the last episode" as you would a standard superlative adjective, meaning either the most recent prior episode or the final episode - the ambiguity comes from the word "last" which has both meanings (it has nothing to do with TV series). Superlatives always require the definite article. You cannot say "a last episode" for the same reason you cannot say, "I am a best student."
Beyond that, the standard rules for articles with nouns and noun phrases apply. Hopefully these rules are clear to you. Here are some examples, in case they're not:
Best episode is the one where Kramer gets a hot tub -> The best episode is the one where Kramer gets a hot tub.
During long episode, I have to take a break -> During a long episode, I have to take a break.
In last episode, John met with Karen. -> In the last episode, John met with Karen.