I think I found an answer. All you need to do in Kernel K means is to compute
$$
C^{(t+1)}(i) = argmin_k \{K(x_i,x_i) -\frac{2}{N_k}{\Sigma_{l\epsilon C^{t}_k}}K(x_i,x_l) +\frac{1}{N_k^2} {\Sigma_{{l,{l`}}\epsilon C^{t}_k} }K(x_l,x_{l`})\} ...(1)
$$
So this is the only operation that needs to be done. One need not to know each cluster center in high dimensional space. Just compute $(1)$ again and again till the algorithm converges.
Algorithm:
Step 1: Assign Random Cluster to points (Known as clsuter map $ C(i):= \{k: i\rightarrow k\}$ i.e point $i$ is assigned to cluster $k$
Step 2: For each point perform $(1)$ above and assign new $C(i)$.
Just to be more clear at this step:
$\rightarrow$After running this step for $(t-1)^{th} iteration $, you get a new $C^{(t)}(i)$ which will be used in (1) again to calculate $C^{(t+1)}(i)$
$\rightarrow$ So each iteration assigns new $C(i)$.Hence, $C^{(t)}(i)$ keeps on changing ( which is representative of cluster means).
Step 3: Repeat 2 above till the point assignments do not change or any of your error metric is stable.
(I am not sure about the error metric that should be used)
New Point:
Each new point will be classified according to $(1)$ above.