How would I design and build a circuit that would generate check bits for 4-bit word?
In this instance, the same circuit should also be used to generate check bits for when you read data back in case of no error, single bit error, or double bit error.
Furthermore, how would I design a circuit that is capable of detecting and correcting a single bit error, in order to detect double bit and to recognize if it is the case of a single bit error, double bit error, or no error?
How would I correct single error in syndrome C4, C2, C1?
Well there are a lot of ways to generate check bits for a four bit word, but if you are in fact(?) interested in the Hamming code, it sounds like the length 7, dimension 4, distance 3 Hamming code fits your description.
There is a generator matrix in the wiki article from which you can deduce the encoding circuit design, and also a corresponding parity check matrix from which you can deduce the error detection/correction circuit.