How exactly did you take the backup; did you use the EXP utility or did you just keep all the data files?
If the former then you may be able to import though 10g IMP although iirc it's only recommended to IMP files generated with the same version of EXP.
If the latter then sorry you have no chance. You will have to reinstall 9i, restore the data files, then export the data.
If neither and you used the Oracle backup utility, this almost certainly won't work. Backup/restore is designed for backup and restore, NOT migration from one version to the next.
Your post is slightly confusing too... You backed up the 9i database, then removed 9i database altogether and installed 10g client. You cannot import data files into a client-only installation, because there's no database present.