Okay so I can confirm that this acvhievement popped after befriending my 60th character, but note that since Clyde unfriends you early on your count should say 59 when it pops (or 58 if you've already unfriended Al Gore), which means the total will be 120. However, I found the following list that gives 110 lines and I know the Christmas Critters give you 12 friends... so I'm not sure how that works out, exactly.
Important Tip: You need to talk to Clyde right away upon starting. He unfriends you only minutes into the game and you will not be able to get him again!
1. Al Gore – Talk to him between Tweek Coffee & the movie theater.
2. Allie Nelson – Talk to her after you recruit the girls.
3. Amber – Rescue her in the sewer.
4. Annie – Save her from the bullies outside City Hall
5. Bartender – Same as Skeeter.
6. Bebe – Main questline.
7. Big Gay Al – Answer the phone.
8. Bill – Save him during main questline.
9. Billy – Hide ‘N’ Seek sidequest.
10. Bishop of Banff – Main questline. (May have to spare his life.)
11. Bradley – Talk to him after you get Shrink.
12. Butters – Start of Game
13. Cartman – Main quest.
14. Catatafish – Main questline.
15. Chinpokomon Toy Corporation – Collect all 30 using our detailed South Park: The Stick of Truth Chinpokomon Locations Guide.
16?. Chistmas Critters – You find them in the Lost Forest. Starting from the ranch go right, up, right, down, and right again. Accept Satan as your lord and they will all add you as a friend. (This gives you 12 friends, so I'm not sure how this calculates exactly, as that would put you over the 120 mathematically required, but if anybody figures it out let me know.)
17. Chris Donnely – Talk to him after you finish main quest up to recruiting the girls.
18. Clyde – At the very very start of the game you have to befriend Clyde before you talk to Cartman. Clyde unfriends you, so he won’t count in the list of 120 (that’s why the list is only 110 long so far). But he might count as part of the friends list, meaning you could have messed up in the first 30 seconds of the game.
19. Cornwallis – Rescue him in the sewer.
20. Crab Person – Talk to him in the sewer after you get Nagasaki fart attack from Terrance & Phillip in Canada. Go down the manhole cover in the center of town (next to the “Men Working” sign) to take you into the sewer where the Crab Person lives. Follow the path and a ladder down until you reach a big rock blocking the way, use your Nagasaki parting farting ability to blow up the rock and reach the Crab Person to befriend it.
21. Craig – Main story
22. Dad – Talk to him after you get 50 friends.
23. Damien – Inside the Theatre.
24. DogPoo – Talk to him at Kyle’s house. Finish sidequest.
25. Dougie – Outside Butters’ house.
26. Douglas – Pink building between Unplanned Parenthood and Photo Dojo (will need the teleporting probe).
27. Duke of Vancouver – Main questline.
28. Earl of Winnipeg – Main questline.
29. Esther – Talk to her in the News Building
30. Filmore – Hide ‘N’ Seek sidequest. (Last one I found. Hardest to find IMO.)
31. Firkle – Main quest.
32. Flora – Hide ‘N’ Seek sidequest.
33. Fossey – Same as Bill.
34. Francis – Break his snowman and talk to him.
35. Frog King – Main questline.
36. General Disarray – Talk to him in Butters’ secret lair.
37. Heidi – Talk to her after you recruit the girls.
38. Henrietta – Main quest.
39. Ike – Talk to him during assault on Craig’s Keep.
40. Jason – Talk to him at Kyle’s house.
41. Jenny – Hide ‘N’ Seek sidequest.
42. Jessie – Talk to her after you recruit the girls.
43. Jessie Rodriguez – Main questline.
44. Jesus – Priest Maxi sidequest.
45. Jimbo – Finish his sidequest.
46. Jimmy – Talk to him at Kyle’s house. Finish sidequest.
47. Karen – In Kenny’s house.
48. Kelly Gardner – Talk to her on the way west from your house.
49. Kelly Rutherford – Talk to her once you have 30 friends.
50. Kenny – Same as Scott.
51. Kevin Stoley – Finish his side quest.
52. Kyle – Main quest.
53. Lemmiwinks – Grade 5 classroom.
54. Leroy Mullens – On top of News Building after you get shrink.
55. Lola – Talk to her in Post Office after 40 friends.
56. Mayor McDaniels – Finish her sidequest.
57. Micheal – Main quest.
58. Millie – Talk to her behind Unplanned Parenthood.
59. Minister of Montreal – Main questline.
60. Mom – Talk to her in your house.
61. Monica – Main questline.
62. Mr. Adler – PTA member after you get the evidence.
63. Mr. Broflovski – Outside Kyle’s house.
64. Mr. Garrison – PTA member after you get the evidence.
65. Mr. Hankey(Summon) – Finish his sidequest.
66. Mr. Hat – Main questline.
67. Mr. Kim(Summon) – Finish his sidequest.
68. Mr. Mackey – Finish his sidequest.
69. Mr. McCormick – Find him outside the Bar.
70. Mr. Slave(Summon) – Finish his sidequest.
71. Mr. Stotch – Same as Mrs.
72. Mr. Tweak – Buy an Ultra Speed Potion.
73. Mrs. Biggle – Talk to her on the bench outside Post Office.
74. Mrs. Broflovski – PTA member after you get the evidence.
75. Mrs. Cartman – Talk to her on the way out.
76. Mrs. Hankey – Same time at Mr Hankey.
77. Mrs. Marsh – Buy something from her (Tom's Rhinoplasty).
78. Mrs. McCormick – In Kenny’s house after the Meth Tweekers mission.
79. Mrs. Stotch – Inside Butters’ house.
80. Mrs. Tweak – Same as Mr. Tweak.
81. Ned – Same time as Jimbo.
82. Nelly – Talk to her after you recruit the girls.
83. Officer Barbrady – Talk to him outside Tom’s Rhinoplasty
84. Pete – 2nd floor of garage 2210
85. Pete – Main quest.
86. Phillip – Main questline.
87. Polly Prissypant – In Cartman's room, must have Cartman as your buddy to activate.
88. Priest Maxi – Finish his sidequest.
89. Prince of Canada – Main questline.
90. Princess of Canada – Main questline.
91. Principal Victoria – PTA member after you get the evidence.
92. Quaid – Hide ‘N’ Seek sidequest.
93. Randy – Main story
94. Red – Talk to her after you recruit the girls.
95. Romper Stomper – Police Station, break him out.
96. Sally – Hide ‘N’ Seek sidequest.
97. Santa – On the roof of the Police Station.
98. Scott Malkinson – Talk to him at Cartman’s
99. Sergeant Yates – Complete his sidequest once Nazi Zombies are in town.
100. Shelly Marsh – Talk to her after Stan’s sidequest.
101. Simon – Rescue him in the sewer.
102. Skeeter – Finish his sidequest.
103. Sparrow Prince – Main questline.
104. Stan – Talk to him at Kyle’s house. Finish sidequest.
105. Terrance – Main questline.
106. Timmy – Unlock all waypoints.
107. Token – Part of the main quest.
108. Tweek – Part of the main quest.
109. Underpants Gnome – After the Underpants Gnome main story mission, talk to the one in your room and he'll give you a sidequest. Finish that and talk to him to friend him.
110. Wendy – Main questline.