How can a lattice attack be applied to ECDSA signatures? The aim is to check if it is possible to break the ECDSA cryptosystem under the following criteria.  


Suppose that each ECDSA signature is generated by using the GLV method for point multiplication (the random nonce $k$ used in each signature is decomposed into $k_1$and $k_2$). 
For every ECDSA signature, I can obtain few bits (least significant bits, most significant bits, ...) of the respective $k_1$ and $k_2$. 


Is it possible to carry out a lattice attack to gain the value of the underlying secret key ($\alpha$)? If yes, then can you explain the attack in detail?  

PS: I have a beginner level understanding of ECDSA and lattices.
Sure you can do. There are many lattice attacks, using your second assumption, to ECDSA (which also applied to DSA). For instance see  Smart and Howgrave-Graham and Shparlinski and Nguyen. All the lattice attacks base on finding small solutions (for the ephemeral key $k$ and the private key $a$) to the signing equation $sk-ra\equiv H(m)\pmod q.$ If you have enough signatures (i.e. enough $k_i$) then you construct a linear system $\bmod q$ and then a suitable lattice and if you have enough bits from every $k_i$ and $a$ and enough messages you can solve the CVP problem. Also you can apply Coppersmith method to the signing equation.