Janus transition metal dichalcogenide monolayers, characterized by antisymmetric crystal structures and unique physical properties, show great potential for applications in micro/nano-electronic devices and thermoelectrics. In this work, the strain-tuned phonon thermal transport and thermoelectric performance of six Janus transition metal dichalcogenide monolayers are systematically investigated by first-principles calculations. This study focuses on monolayers of PtSSe and PtTeSe with a 1T-phase crystal structure, as well as monolayers of MoSSe, MoTeSe, WSSe, and WTeSe with a 1H-phase crystal structure. All first-principles calculations are performed using the open-source software Quantum ESPRESSO. The lattice thermal conductivity is obtained based on lattice dynamics and iterative solutions of the Boltzmann Transport Equation. The thermal conductivities of PtSSe, MoSSe, and WSSe monolayers are generally higher than those of PtTeSe, MoTeSe, and WTeSe. Acoustic phonons are responsible for the majority of thermal transport, contributing over 95%. Under unstrained conditions, monolayer PtSSe demonstrates a superior thermal conductivity of 104 W·m^-1·K^-1, making it advantageous for thermal management applications in electronic devices. Under tensile strain, the thermal conductivity of PtSSe, MoSSe, and WSSe monolayers exhibits a monotonic decrease; however, for PtTeSe, MoTeSe, and WTeSe monolayers, thermal conductivity initially shows an increase, followed by a subsequent decrease. Under a 10% tensile strain, the thermal conductivities of these six Janus monolayers all demonstrate a reduction exceeding 60%. Furthermore, this work provides a comprehensive analysis of the impact of strain on specific heat capacity, phonon group velocity, and phonon lifetime. Phonon mode-level analysis and cross-calculated thermal conductivity (with specific heat capacity, phonon group velocity, and phonon lifetime replaced by values under different strain conditions) reveal that phonon lifetime is the dominant factor governing thermal conductivity under strain. For electrical transport properties, calculations are performed using the Boltzmann transport equation based on deformation potential theory. At room temperature, the thermoelectric figure of merit (ZT) for PtTeSe is 0.91 without strain, which can be improved to 1.31 under 10% tensile strain. The ZT value reaches as high as 3.96 for p-type PtTeSe and 2.38 for n-type PtTeSe at 700 K, indicating that PtTeSe monolayer is a highly promising thermoelectric material. Strain-induced enhancements in the thermoelectric performance of PtTeSe are facilitated by a reduction in lattice thermal conductivity and a reconfiguration of the band structure. This work demonstrates that strain engineering is an effective strategy for tuning the thermal transport and thermoelectric performances of Janus transition metal dichalcogenide monolayers.