The deployment of small cells (SCs) in heterogeneous cellular networks (HCNs) has been shown to be a promising approach to increase the system capacities of wireless communications. However, the bandwidth sharing and densification of the SCs cause co-tier and cross-tier interferences that limit the system spectral efficiency. In this paper, a framework of an uplink area spectral efficiency (ASE) analysis for multichannel HCNs using a proposed biased cell association scheme with coordinated subchannel allocation and channel inversion power control for mitigating the interferences is developed. The coordinated subchannel allocation is proposed to alleviate the severe cross-tier interference caused by macrocell users around the edges of the SCs. By incorporating the coordinated subchannel allocation into the biased cell association scheme with a bias factor, the proposed scheme is able to design the bias factor to reshape the coverage areas of the SCs providing an effective tradeoff between user offloading and cross-tier interference avoidance. By applying the tools of stochastic geometry, the uplink ASE is analytically derived as a function of the bias factor for the multichannel HCNs. The optimal bias factor can be obtained numerically by maximizing the system ASE. It is shown that the ASE performance analysis agrees with computer simulation very well. The results demonstrate that the proposed scheme can substantially improve the ASE performance of the multichannel HCNs for different small cell densities and outperform upon an existing biased cell association scheme.