The title says it. I am looking for a good exposition on the Ktheory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$. I have some background in $K$theory and also some background in modular curves.

$\begingroup$ Maybe you should specify a little more explicitly what you are looking for. Beilinson's conjectures give a fairly general picture of what is expected of the Ktheory (in particular of modular curves). Besides Beilinson's papers on higher regulators for modular curves, there are a couple of expository papers on Beilinson's conjectures by Nekovar, DeningerScholl, SchappacherScholl, etc. There is quite a followup literature on construction of special elements in Kgroups, but essentially Beilinson's conjectures remain unproved even in the curve case. $\endgroup$ – Matthias Wendt Dec 21 '15 at 15:31

1$\begingroup$ Beilinson's "Higher regulators and values of Lfunctions", J. Sov. Math. 30 (1985) is supposed to be a good (general) reference here. $\endgroup$ – jvo Dec 21 '15 at 20:56
I wouldn't recommend Beilinson's 1985 paper as a general reference  it's terrifyingly compressed, developing an entire new subject in a single short paper, and crashes through the necessary material on modular curves in a couple of sentences.
A much gentler reference would be Flach's 1992 Inventiones paper "A finiteness theorem for the symmetric square of an elliptic curve", which gives a much fuller introduction to Kgroups, Gersten complexes, regulator maps etc. You could also try looking at parts of my paper with Lei and Zerbes "Euler systems for RankinSelberg convolutions" for more on this.