Become a Legendary clan or Become an Official clan.
Both are pretty self explanatory.
Desc for Legendary should be, "Your clan has become.... LEGENDARY!"
Desc for Official should be, "Your clan has grown up! Relish in your achievements!"

Neither of those are in the achievements list?

Not as far as I can see. Here's all the achievements available.

...but we're asking for specific achievements' descriptions, not all currently availble and added already?

oh shit. I'm sorry. I didn't read too good. I thought it was asking for new achievements.
In that case, for clan streak:
Tier 1, "Break down the competition!".
Tier 2, "Let no others stand in your path!".
Tier 3, "Amazing! You beat the competition into the ground!"

Clan level achievements(20 total)
Level 5
Nice work fresh bloods!
Level 10
No longer avaible more fresh bloods in this clan!
Level 15
Beginner nerds!
Level 20
Clan work:XP hunting!
Level 25
The uncrowned kings!
Level 30
For pro players only!
Level 35
Nice work you guys are advanced nerds now!

Clan Wars:
1: Already done. "You have won your first clan war!"
5: "Eh, not bad. You've won 5 wars."
10: "Slowly getting there. You've won 10 wars."
25: "On the rise! You've won 25 wars!"
50: "Bodacious man! You've won 50 wars!
100: "Truly admirable! You've won 100 wars!
---
Added 500, 1000. Long term goals only the most dedicated will achieve. WAPOW/Obey have 500+ won.

500: "Woah... what? You've won 500 wars!!"
1000: "The best ever? INCREDIBLE! You've won 1000 wars!!!"

Feel free to expand the contracted 'you've' if you wish.
-------------------
Levels:
1: "Your way has started." <---- add a period, currently lacking
5: "Baby steps lead to giant strides."
10: "Time to become official."
15: "Training wheels come off now."
20: "Getting better... Harder... Stronger!"
25: "Well done, progress has been made."
30: "Pedal to the metal, floor it!"
35: "A third of the way to your final goal."
40: "Not too shabby, keep it up."
45: "You can't stop now - Nearly lvl 50!"
50: "Halfway to glory!"
55: "Keep on keeping on, never give up."
60: "Look at how you've come. Few clans reach this point."
65: "Whose bright idea was it to have new descriptions written every 5 levels?"/"Left, right, left, right! Keep marching soldier!"
70: "Other clans envy your mad XP total!"
75: "Time for the final stretch!"
80: "Welcome to the grind."
85: "if ur reading dis, your clan is thicc af"
90: "So close, yet still so far.."
95: "Your clan is the 1%."
100: "Hallelujah!!! Your clan has achieved holy status! You are gods amongst men!"
-----
/me exhales
phew...

Level

Level 5: "Keeping up the show! "
Level 10: "In coming!"
level 15: "You're growing like a teen."
level 35: "Another cup of XP, please!"
level 50: "Man in the middle."


Win streak


3: "T-T-Take that!"
10: "No mercy"

Endurance

1 month: "Young blood"
3 month: "Hey! We're still here."
6 month: "Looks like you came to stay"

Clan Wars


5: "You won 5 wars, keep it up!"
10: "Ten is better than five"
50: "Up to the glory!"

Breaking war streak

5: "What a bad boys!"
10: "Omae wa mou shindeiru"

Win 50 clan wars
Desc: Your clan has reached 50, looks like there are still many more clans lurking!

You have started your clan
Desc: Bring it on!

Sweet Victory!
Desc: Become #1 Clan!

Still Fighting
Desc: Win 10 wars

Become Official
Desc: You have reached fame

Why Haven't you lost yet?!?
Desc: win 3 wars in a row

Level 1
Desc: You've started your path to greatness

Loosen up and get ready to fight
desc: Start your first war