Name That STAR TREK Episode...

Okay, new clue (Hint - all the clues are about one person):
5th Clue: A dual personality.

 

The Alternative Factor?

 

You are in the wrong universe. Lazarus didn't have a dual personality; he/they were two separate people. Please try again.

Summary (Hint - all the clues are about one person):
1st Clue: Blame Starfleet for the problem.
2nd Clue: A little hot under the collar.
3rd Clue: A little transparent.
4th Clue: Not dead, yet.
5th Clue: A dual personality.
6th Clue: <Sniff.> Maybe I really need to freshen up.

 

The Enemy Within?

 

Like Lazarus, Kirk didn't have a dual personality; they were two separate people. There are several other clues...Please try again.

7th Clue: First I'm disgusting, then I'm beautiful. Half of me likes it.

 

Turnabout Intruder?

 

The answer must be either black or white, so here goes nothing...

 

No to "both" of you. (pun intended). New clue:

1st Clue: Blame Starfleet for the problem.
2nd Clue: A little hot under the collar.
3rd Clue: A little transparent.
4th Clue: Not dead, yet.
5th Clue: A dual personality.
6th Clue: <Sniff.> Maybe I really need to freshen up.
7th Clue: First I'm disgusting, then I'm beautiful. Half of me likes it.
8th Clue: I'll have love and companionship until I die. (I running out of clues...)

 

MUDD'S WOMEN?

Eve is disgusted by the use of the drug.

 

You're close, but not correct. It's amazing on how many of the clues can apply to unintended other characters. Sorry, keep searching for those lithium crystals. (Hint: the last clue really tells it all.)

 

THE NAKED TIME?

 

You must be affected by the planet's complex water molecule chain to stubble past the obvious clue. I feel like waiting for the correct answer is like waiting for Scotty to finish cutting though that wall...

 

Metamorphosis

 

YES, thank you @Odona Kirk. It's Metamorphosis.
Hint 1: all the clues are about one person.
All clues directly involve Assistant Federation Commissioner Nancy Hedford.
Hint 2: the last clue really tells it all.
"Companion" is given in the last clue.
1st Clue: Blame Starfleet for the problem.

NANCY: Small thanks to the Starfleet.
MCCOY: Now really, Commissioner, you can't blame the Starfleet.
NANCY: I should've received the proper inoculations ahead of time.​

2nd Clue: A little hot under the collar.
McCoy was worried that she would develop a fever as the first signs of the disease. During the episode, she indeed develops said fever. The clue also refers to her continued anger during the episode.
3rd Clue: A little transparent.
Reference to both the Companion's appearance, and later when she looks though her transparent scarf.
4th Clue: Not dead, yet.
On her death bed, the companion merged with Nancy which brings her to perfect health.
5th Clue: A dual personality.
After the merger, both Nancy and the Companion are inside her body/brain.
6th Clue: <Sniff.> Maybe I really need to freshen up.

COCHRANE: Marooned, I told you. Look, we'll have lots of time to learn about each other. I have a small place over that way. All the comforts of home. I can even offer you a hot bath.
NANCY: How perceptive of you to notice I needed one. ​

7th Clue: First I'm disgusting, then I'm beautiful. Half of me likes it.
When Cochrane learns that the Companion is feeding off him in a lover relationship, he says:

COCHRANE: That thing fed on me. It used me. It's disgusting.​

Later after the merger, their conversion goes as:

COCHRANE: You're very beautiful.
NANCY: Part of me understands. Part does not. But it pleases me.​

8th Clue: I'll have love and companionship until I die. (I running out of clues...)
Condensing the following dialog:

SPOCK: But you will age, both of you. There will be no immortality. You'll both grow old here and finally die.
COCHRANE: That's been happening to men and women for a long time. I've got the feeling it's one of the pleasanter things about being human, as long as you grow old together.
KIRK: Are you sure?
COCHRANE: There's plenty of water here. The climate's good for growing things. I might try to plant a fig tree. A man's entitled to that, isn't he? It isn't gratitude, Captain. Now that I see her, touch her, I know that I love her. We'll have a lot of years together. They'll be happy ones.​

And last, I put reference to the Companion in the clue, which I hoped would trigger the correct answer so I can pass the game on. Whew.

 

Thank you @Henoch. Clue 1: Bad math.

 

Devil In the Dark? Spock gives an "approximation" of their odds of being both killed.

 

No, sorry.

 

The problem is that I don't think much of their maths in general and I, therefore, don't pay any attention to the numbers they give.

 

Second clue: Bad chess.

 

Court Martial.

1. There's the infamous reference to "one to the 12th power" or whatever it is.
2. The computer plays chess poorly, allowing Spock to beat it, and giving everyone the needed "AHHHHHHHHH" moment to realize that someone (FINNEY!!) tampered with the Enterprise's computers.