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I am testing the security of a particular functionality of the system and exploring ways it can be hacked.
My question is:
If suppose I have a string ABCDE and I encrypt it using a key. The encrypted string is VWXYZ. Now if I have such several combinations. That is to say I have a lot of encrypted data and its decrypted form. Can I somehow get the encryption algorithm?
No. You cannot "get" the encryption algorithm.
(By this, I'm assuming you are asking if you can prove which encryption algorithm is being used.)
The best you can say is that, based on the combinations you have tried, the encryption algorithm appears to be X.