Will I be able to depreciate the "leasehold improvement" portion of a business purchase and if so, for what time frame?

Unfortunately, an improvement made by the lessor does not qualify as qualified leasehold improvement property to any subsequent owner unless it is acquired from the original lessor by reason of the lessor's death or in one of the types of transactions listed in the reference below.

See the specific rules under Qualified Leasehold Property (IRS Pub 946): Which Property Class Applies?

Thanks, but the lessee made the improvements and that is who I am purchasing the business from. Does that make a difference or does the same rules apply?

You are the subsequent owner, so the rule would apply. The leasehold improvements are not depreciable property for you. The total cost would be your basis in the business (also not subject to depreciation).