Is it possible to encrypt the key by sender's private key and again encrypt it with receiver's public key by using RSA algorithm at sender end Is it possible to encrypt the data by sender's private key and again encrypt it with receiver's public key by using RSA algorithm at sender end.How decryption is done at receiver's end?
It doesn't make much sense to "encrypt" with your private key, I suspect what you want to do is sign with a private key, which for RSA is essentially the same operation. 
You definitely can sign a message with your private RSA key and then encrypt it with the recipients RSA public key. This is even common. 
Obviously you want to use proper padding (e.g OAEP)

Note you can do this in either order, sign and encrypt or encrypt and sign. 
Note public key operations are cheaper computationally compared to private key operations. If you encrypt and sign we can verify cheaply and not waste time decrypting unverified messages. If you sign and encrypt you can hide you identity If this is relevant.