What design should I use?
If you want to have a computer be assigned to more than one user, you're going to need a junction table between the two to store the relationships. This was not stated in the requirements but it seems like a logical approach given the project scope.  1. Account 2. Computers 3. Account_Computer_Assignments.  The third one is your junction table that stores the relationship between the other two.  It's primary key would consist of IDAccounts and IDComputers using your naming convention.  A foreign key relating back to their parent tables would also be advisable.
This structure will also allow for multiple users to link to the same computer.  I don't know if that is the intended behavior but it will allow for that with this design.
If that is not intended, and a single computer can only be monitored by one and only one user, then a single field on the Computer table for the AccountID is all that is needed.  No junction table.