The DES (Data Encryption Standard) takes 64 bit key as input but how is the plain text of length say n is getting converted into 64 bit binary format?
The conversion to binary is not part of the DES standard. So you'll have to define that part of the protocol yourself. The cipher itself needs to be used with a mode of operation, as fgrieu indicated.
There are basically two families of modes of operation: streaming modes of operation and block modes of operation.
In block modes sych as ECB or CBC you have to split the plaintext into separate, sequential blocks, usually after padding the plaintext.
In streaming modes such as CFB and OFB the bytes may be encrypted byte for byte.
Of course the API (the programming interface) of high end implementations of ciphers or modes may hide such implementation details.