Kalsoom, Kulthum and Kulthoom are all correct spellings. The name is originally pronounced as [(C)ute + p(U)t + p(L)ay] + [(TH)ing + s(OO)n + (M)e], which can be written as Kulthum, Kulthoom, Kolthum, Kolthoom, Kulthume, Kulthoum, and many other ways.
Most people, including most Arabs, change the th sound to s, since s is easier to pronounce. Thus they pronounce it as [(C)ute + p(U)t + p(L)ay] + [(SOO)n + (M)e], which can be written as Kulsum, Kulsoom, Kolsum, Kolsoom, Kulsume, Kulsoum, and others.
Additionally, some people pronounce the beginning with an a sound, as in [(CU)p + p(L)ay] + [(TH)ing + s(OO)n + (M)e], which can be written as Kalthum, Kalthoom, Kalthume, Kalthoum, or with the s sound, as in Kalsum, Kalsoom, Kalsume, Kalsoum, etc.
All of the pronunciations mentioned above are correct and acceptable. You can choose any spelling you like, or make your own spelling.
For the meaning of Kalsoom please see this answer: What do Zarafshan and Kulsoom mean?