Some online backup providers have been known to slow down your computer by using up extensive resources when running in the background. Some have also been known to slow down your internet connection when backing up or restoring.
If we experienced this with any providers we have documented this in our reviews. We recommend you read customer comments and feedback on provider reviews to check for this as this is the biggest customer complaint for online backup.
If you go with a top rated provider like JustCloud, Carbonite, Mozy, MyPCbackup, ZipCloud, BackupGenie, Sugarsync, Dropbox you should be ok as these are all known to consistently run fast and have good quality applications.
Age of your computer, memory etc will also affect this, so like any application you will be using some resources but this should be mimimal, main speed issues will be the backup.
Remember though, backing up data is seen as uploading not downloading. This is notoriously slower and initial backups have been known with every provier to takes days and sometimes weeks. Complete computer backups are rarely done in just 1 day.