Proudmoore Admiralty

If your Kul Tiran is still locked, even after you've completed all achievements, make sure the following quest chain is complete as well:

1. 
   
2. Head to Stormwind.
3. 
   
4. Fly north to the Embassy of Stormwind.
5. 
   
6. Accept Made in Kul Tiras from Jaina (main quest that starts a long a** chain)
7. 
   
8. Port/Hearth to Boralus.
9. 
   
10. Go to the Harbormaster's Office and speak with Cyrus Crestfall. Tell him "I am told you know where to find the shipwright, Dorian Atwater?"
11. 
    
12. Listen to the dialog.
13. 
    
14. Turn in the quest and accept The Master Shipwright.
15. 
    
16. Speak with Jaina and <Take the letter.>
17. 
    
18. Fly to Brennadam, Stormsong Valley.
19. 
    
20. Speak to Cyrus to turn in quest.
21. 
    
22. Accept Fish Tales and Distant Sails.
23. 
    
24. Speak with Talkative Brennadam Citizens and ask them if they know where to find Dorian.
25. 
    
26. Once you learn Dorian's location, go back and talk to Cyrus to turn in the quest.
27. 
    
28. Accept Home, Home On the Range.
29. 
    
30. Fly to Deadwash, Stormsong Valley.
31. 
    
32. Ride NW to /way 49.94 50.04
33. 
    
34. Talk to Old Atwater and ask him if he is Dorian Atwater.
35. 
    
36. Go inside the house and turn in quest to Dorian.
37. 
    
38. Accept I'm Too Old for This Ship.
39. 
    
40. Go outside the house and saddle the horse Hurricane (it may take a few tries).
41. 
    
42. Use your Flight Master's Whistle to get to the nearest flight path.
43. 
    
44. Fly to Tidecross, Stormsong Valley.
45. 
    
46. Speak to Dorian to trun in quest.
47. 
    
48. Accept Covering Our Masts and The Deep Ones.
49. 
    
50. Slay 12 Drowned Forces while clicking on all 4 Fog Wards to gain Concealing Fog (it will make you less detectable to enemies)
51. 
    
52. Go back to Dorian and turn in quests.
53. 
    
54. Accept Frame Work.
55. 
    
56. Port/hearth to Boralus.
57. 
    
58. Fly to Heatherford, Tiragarde Sound.
59. 
    
60. Ride NW to /way 68.86 20.44 and enter the buidling
61. 
    
62. Speak to Dorian and turn in quest.
63. 
    
64. Accept Team Carry.
65. 
    
66. Collect 4 Heavy Lumber then follow Dorian on the path heading north (she's so annoying ugh)
67. 
    
68. Speak to her to turn in quest.
69. 
    
70. Accept The Bleak Hills.
71. 
    
72. Use your Flight Master's Whistle to get to the nearest flight path.
73. 
    
74. Fly to Fletcher's Hollow, Drustvar.
75. 
    
76. Ride to /way 61.90 59.33 and speak with Thornspeaker Birchgrove to turn in quest.
77. 
    
78. Accept Gorak Tul's Influence.
79. 
    
80. Fill the Drust Vessel by killing mobs and gathering Empowered Minor Cairns (they look like tall rocks with an arrow pointing down).
81. 
    
82. Go back to Thornspeaker Birchgrove to turn in quest.
83. 
    
84. Accept Balance in All Things.
85. 
    
86. Talk to Arthur Tradewind (druid cat next to you) and ask him to take you to the High Thornspeaker. He will shapeshift to a druid stag for you to ride.
87. 
    
88. Speak with Ulfar and ask him to tell you how you can help.
89. 
    
90. Click the Ancient Seedling to begin the ritual. Run over the green orbs and kill mobs that spawn to complete the ritual.
91. 
    
92. Talk to Ulfar and turn in quest. Go outside the cave and use your Flight Master's Whistle to get to the nearest flight path.
93. 
    
94. Fly to Watchman's Rise, Drustvar.
95. 
    
96. Go west to /way 33.15 30.38 and speak with Chelsea Wright to turn in quest.
97. 
    
98. Accept Drop It!
99. 
    
100. Follow the Wicker Beast that spawned next to you. He will guide you to the cave at /way 36.31 29.25
101. 
     
102. Click the Storeroom Keys. Talk to Wicker Beat to turn in quest.
103. 
     
104. Accept Her Dog Days Are Over.
105. 
     
106. Jump down to /way 35.33 28.93 and click Gnawed Bone.
107. 
     
108. Ride to /way 35.99 30.94 and click Worn Collar.
109. 
     
110. Go inside the house at /way 34.59 30.59 and click Squeaky Toy.
111. 
     
112. Go back to /way 33.15 30.38 and speak with Chelsea Wright to turn in quest.
113. 
     
114. Accept Make it Wright.
115. 
     
116. Click the Storage Crate next to you to turn in quest.
117. 
     
118. Speak with Bound Sky next to you.
119. 
     
120. Accept Summons from Dorian.
121. 
     
122. Hearth/port to Boralus.
123. 
     
124. Fly to Tidecross, Stormsong Valley.
125. 
     
126. Ride to /way 66.51 44.23 and talk to Dorian to turn in quest.
127. 
     
128. Speak to Jaina next to you and accept A Worthy Crew.
129. 
     
130. Hearth/port to Boralus again (my bracers have Cooled Hearthing enchant) or use your Flight Master's Whistle to get to the nearest flight path.
131. 
     
132. Fly to Proudmoore Keep, Tiragarde Sound.
133. 
     
134. Go upstairs to /way 42.95 56.25 and speak with Tandred Proudmoore and tell him "Jaina has sent me to spak to you".
135. 
     
136. Hearth/port to Boralus again.
137. 
     
138. Go to /way 75.94 23.01 and speak with Brother Pike. Ask him if he's interested.
139. 
     
140. Fly to Vigil Hill, Tiragarde Sound.
141. 
     
142. Go to /way 56.68 61.17 and talk to Lieutenant Tarenfold. Ask him if he's interested.
143. 
     
144. Go back to the flight master and fly to Outrigger Post, Tiragarde Sound.
145. 
     
146. Recruit Rosaline Madison at /way 35.26 24.14.
147. 
     
148. Hearth/port to Boralus.
149. 
     
150. Fly to Tidecross, Stormsong Valley.
151. 
     
152. Go to /way 67.63 44.90 and speak with Dorian. Tell her "Well, you heard the man".
153. 
     
154. Talk to Jaina next to you and turn in quest.
155. 
     
156. Accept Blessing of the Tides.
157. 
     
158. Talk to Jaina again and choose a name for the vessel out of the 4 choices. (I chose Dawnsailor)
159. 
     
160. Follow Brother Pike and click on the Rowboat. (watch/skip cutscene)
161. 
     
162. Talk to Jaina and turn in quest.
163. 
     
164. Accept Allegiance of Kul Tiras.
165. 
     
166. Click the Captain's Helm (in front of you) to sail to Stormwind.
167. 
     
168. Jump down on the deck and speak with Anduin to turn in quest.
169. 
     
170. Achievment Earned: Allied Races: Kul Tiran
171. 
     

Congratulations, now go create your Kul Tiran!

As of the current patch in Shadowlands, you no longer need to be exalted with Proudmoore Admiralty in order to complete this achievement. Source: I am 3k into Honored with Proudmoore Admiralty and just finished this quest chain and completed this achievement.

Starting from patch 10.1.5, all restrictions for unlocking Allied Races have been removed, with the exception of a level requirement.
All Allied Races can now be unlocked once you have a level 40 character on your account (was level 50). Blue post.