Realizing high-rate and high-capacity features of Lihium-organic batteries is essential for their practical use but remains a big challenge, which is due to the instrinsic poor conductivity, limited redox kinetics and low utility of organic electrode mateials. This work presents a well-designed donor-acceptor Covalent Organic Framework (COFs) with extended conjugation, mesoscale porosity, and dual redox-active centers to promote fast charge transfer and multi-electron processes. As anticipated, the prepared cathode with benzo [1,2-b:3,4-b′:5,6-b′′] trithiophene (BTT) as p-type and pyrene-4,5,9,10-tetraone (PTO) as n-type material (BTT-PTO-COF) delivers impressive specific capacity (218 mAh g⁻¹ at 0.2 A g⁻¹ in ether-based electrolyte and 275 mAh g⁻¹ at 0.2 A g⁻¹ in carbonate-based electrolyte) and outstanding rate capability (79 mAh g⁻¹ at 50 A g⁻¹ in ether-based electrolyte and 124 mAh g⁻¹ at 10 A g⁻¹ in carbonate-based electrolyte). In addition, the potential of BTT-PTO-COF electrode for prototype batteries has been demonstrated by full cells of dual-ion (FDIBs), which attain comparable electrochemical performances to the half cells. Moreover, mechanism studies combining ex situ characterization and theoratical calculations reveal the efficient dual-ion storage process and facile charge transfer of BTT-PTO-COF. This work not only expands the diversity of redox-active COFs but also provide concept of structure design for high-rate and high-capacity organic electrodes.