What would happen if I own a business with a lease and the city condemned the building, would I be compensated by either my landlord or the city?

If the city condemmed the building because the landlord violated health, safety, building, etc. codes, you would have a potential claim vs. the landlord: in that case, the landlord violated his obligations under the lease to provide you space for the duration of the lease's term. You could sue the landlord for lost income and/or additional costs caused by this, with some prospect or chance of winning. But if the condemndation was not due to the landlord's fault (e.g. the building was condemned due to some change in ordinances, or due to damage, such as from flooding, weather, fire, burst water main, etc., beyond the landlord's control), the landlord would not be liable: the landlord's potential liability is based on the landlord, and not some other party, city government, or natural forces, violating your lease. In this second situation, you would not have a viable case or claim.
The city is not liable except perhaps in the unlikely event you could prove that the city or its employees acted improperly or out of animus: e.g. condemned the building due to a personal dispute or conflict with your landlord, or because the landlord refused to bribe them, or to help out some other business or landlord by eliminating competition, etc. Only if you could prove an intentional wrongful act or act beyond the city's authority might you be able to sue them and win.