You are about to apply for the Civil Services Examinations. Any job application related to the Government service should have clear details regarding your identity (name, date of birth, caste etc), because once you apply, then changing the details is difficult unless you have a valid reason.
Firstly, check all the identify documents that you have and the name in those. Does it match with your new name. If not, then you need to change all of them because, once you clear the UPSC CSE prelims, you will have to undergo the document verification where irregularities will be frowned upon and the job itself would be at risk.
The important documents include
Date of Birth certificate
10th marks card
12th marks card
Bank account name
Do you have legal documents with your new name? Please check all these and MOST IMPORTANT, the last date before which you have to apply for the UPSC exams.
If you do not have time and all the identity documents are in your old name, do not want the miss a chance in UPSC then it would be a good idea to apply first, get the job, join the Government service job and after some months start the process of changing the name.
If you have time and or the new name is very important to you, then you need to meet a Notary or a First Class Judicial Magistrate.
He/she will guide you for getting an affidavit that states you new name. Once this is done, you need to publish it in at least two newspapers, one English, one vernacular or regional language. Once you have done this then you legally go by the new name. Please preserve the affidavit and the original cuttings of the newspaper announcements.
Then you need to start the long process of changing your details in all the Identity documents that you have inclduing the bank accounts. Give it a thought and then discuss with a good lawyer.
Please check this link
in ISC site that has details about how to go about changing details in existing ID documents that I have listed above.