When a piece of DNA carrying a gene of interest is transferred into an alien organism, its fate depends on several factors, including the mechanisms it has for replication, integration, and inheritance. Here is an analysis of each statement provided:

A. The piece of DNA would be able to multiply itself independently in the progeny cells of the organism.
This is incorrect unless the DNA contains an origin of replication that is compatible with the host cell's replication machinery.

B. It may get integrated into the genome of the recipient.
This is a likely scenario if the DNA has sequences that facilitate recombination with the host genome or if the host has natural mechanisms of DNA integration such as those observed in bacteria with transformation capabilities, or other eukaryotic organisms with similar processes.

C. It may multiply and be inherited along with the host DNA.
This statement can be true if the introduced DNA is replicated along with the host DNA, possibly through integration into the host genome or existence in the form of an episome that replicates independently yet synchronously with the host DNA.

D. The alien piece of DNA is not an integral part of chromosome.
This statement is generally accurate before integration. The introduced DNA, before integration, exists extrachromosomally and thus is not part of any chromosome, unless mechanisms are present for its integration into the host genome.

E. It shows ability to replicate.
This statement depends on specific sequences in the DNA, such as the origin of replication compatible with the host cell's machinery, mentioned in statement A. Without these, it will not replicate independently.

Given these explanations, the most accurate answer that describes the fate of the DNA in the recipient organism is:

Option C: B and C only
'B' is correct as the DNA may integrate into the host genome, and 'C' is correct as the DNA might also multiply if replicated in sync with the host organism's DNA, particularly if it integrates or exists as an episome compatible with the host's cellular machinery.