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In this paper, we introduce an $N\times N$ matrix $\epsilon^{a\bar{b}}$ inthe quantum groups $SU_{q}(N)$ to transform the conjugate representation intothe standard form so that we are able to compute the explicit forms of theimportant quantities in the bicovariant differential calculus on $SU_{q}(N)$,such as the $q$-deformed structure constant ${\bf C}_{IJ}^{~K}$ and the$q$-deformed transposition operator $\Lambda$. From the $q$-gauge covariantcondition we define the generalized $q$-deformed Killing form and the $m$-th$q$-deformed Chern class $P_{m}$ for the quantum groups $SU_{q}(N)$. Someuseful relations of the generalized $q$-deformed Killing form are presented. Interms of the $q$-deformed homotopy operator we are able to compute the$q$-deformed Chern-Simons $Q_{2m-1}$ by the condition $dQ_{2m-1}=P_{m}$,Furthermore, the $q$-deformed cocycle hierarchy, the $q$-deformed gaugecovariant Lagrangian, and the $q$-deformed Yang-Mills equation are derived

q-deformed Chern characters for quantum groups SUq(N)