I get that her personality is greedy all powerful dragon lady but still, I need a more deep explanation... and our next session (the one where that would help) is in two weeks AAAAAAA!

 Why are your PCs encountering Tiamat without you having figured this out ahead of time? In any case, Tiamat's personality isn't incredibly well defined, so make up what she's like in your game.

Expanding on the list of personality traits in the FR Wiki, she is:

• Arrogant: Believes herself superior to all, including other gods and dragons.
• Greedy: Obsessed with acquiring wealth, power, and treasures - prefers gifts from followers.
• Hateful & Spiteful: Holds grudges, never forgives a slight, and delights in revenge.
• Vain: Deeply self-absorbed, basks in her own glory, and demands worship and adoration.
• Manipulative: She is cunning and believes mortals are expendable pawns. In fact, she rarely acts directly, preferring to work through agents, cults, and dragons.
• She enjoys the company of chromatic dragons, but treats them as possessions or consorts, not equals.

Her heads have different personalities (red: prideful, blue: arrogant, green: deceitful, black: cruel, white: primal). You can have them speak in unison or in turns, by her mood.

You might want to checkout "Through the Eyes of Tiamat" on Youtube - it gives a take on her personality (not the only one) which might give you some ideas.

Hope it helps.

(oh, and as to Vlad's final comment that her heads have different personalities - I always believed that she had a single personality INCLUDING all these traits (but that might be older lore and you could play it that way if you wanted to.)

Thanks! I already watched Through the eyes of Tiamat. Also, my players don't know yet and I have thought this through (badly)

You can just make up what you need to. Your players aren't going to have a strong idea of what Tiamat's personality is like, and also you're the person in charge of the universe, so can change it according to your needs.

Also, Tiamat is presumably highly intelligent, and can present different aspects of her personality if she wants to, and it matters what context the players are communicating with her. Are they coming to her a supplicants, needing a boon? Does she actually want something from them? 

Ohhhh. Thats a hard one. I assume each head would have a different personality, and they would argue a lot with each other. This would be complicated, but it would add depth to the game.

Yay

Thanks a lot, you will be credited in the acknowledgments of my D&D campaign when it comes out.