I don't know if this has been posted before. Stop me if it has.
With the exception of the six units you must take into the tower, the herons, the three you get inside the tower, and a certain Burger King, there are sixty playable units in FE10. You have free choice of any ten to take into the tower (really, I'm not so mean as to disallow everybody except those ten and any forced units for the entirety of the game, though you're welcome to if you really want). As such, it's easy enough to play six runs, using an entirely different team for the final chapter every time.
As for who you use in which run, you can choose manually if you're frightened of what luck might give you, but I prefer to use the Sequence Generator on random.org. Assign every relevant unit a number between 1 and 60, then choose to generate the numbers from 1 to 60 in a sequence, and for ease on the brain, format it in six columns. There, you have six randomly chosen teams of ten, represented by the numbers assigned. You might want to write these down or something.
Personally I'm doing this on Easy mode just so I get to know what it's like to use each of the characters without any downsides being too extreme or frustrating (read: Lyre). You're welcome to do this on whatever difficulty you want, really.
I'll number each unit so you don't have to:
1. Edward
2. Leonardo
3. Nolan
4. Laura
5. Ilyana
6. Aran
7. Meg
8. Volug
9. Jill
10. Tauroneo
11. Zihark
12. Fiona
13. Tormod
14. Muarim
15. Vika
16. Nailah
17. Elincia
18. Marcia
19. Nealuchi
20. Haar
21. Brom
22. Nephenee
23. Heather
24. Lucia
25. Mordecai
26. Lethe
27. Geoffrey
28. Kieran
29. Astrid
30. Makalov
31. Danved
32. Calill
33. Titania
34. Soren
35. Mist
36. Rolf
37. Boyd
38. Oscar
39. Shinon
40. Gatrie
41. Rhys
42. Mia
43. Ranulf
44. Kyza
45. Lyre
46. Janaff
47. Ulki
48. Sigrun
49. Tanith
50. Skrimir
51. Tibarn
52. Naesala
53. Pelleas
54. Stefan
55. Oliver
56. Bastian
57. Volke
58. Caineghis
59. Giffca
60. Renning
Discussplz.