The first requirement is a VFD that is rated for single-phase, 230 V input and output current and power that is adequate for your motor. The VFD must also be rated for a constant torque load. Some are intended only for fans and centrifugal pumps. Some manufacturers may call VFDs for constant torque loads "heavy duty." They are usually rated to provide 150% of rated output current for one minute.
You may want to consider buying from a local supplier that may be willing and able to answer questions about the use of the product. Consider the local reputation of both the supplier and the manufacturer.
If you absolutely need to disconnect the motor instantly for some safety reason, you need a contactor. Using the VFD to start and stop the motor is best except for that kind of circumstance.
You don't need relays unless there is some interface with the control of the overall machine that requires that.
You do need branch circuit protection. The VFD manual should provide guidance about selecting that. There are other installation details that you should learn about before purchasing the VFD.
Before purchasing the VFD, you should download the manufacturer's instruction material and study it. If that is not available, you probably look for products of other manufacturers.