This is non-trivial design task, a good solution requires knowledge of your system and environment (if the self-test is not just a formality). The functional safety library linked by Jan provides building blocks for many components - in case you are ready to involve their NDAs...
Basically, you should trust the core MCU components: the internal oscillator, (part of) internal SRAM, (part of) internal flash, timers etc. and the power supply. If this is broken, the MCU won't run anything, including the self test. The built-in brownout detector can help with check of the power supply (but this is not software).
Using these resources, check remaining internal components (checksums/signatures in internal flash, other banks of internal RAM), state of well-known GPIOs, analog signals. Then important external components: external oscillator (if present), external memories. Then other complex external components (devices attached to SPI, I2C, and so on). For continuous test: enable runtime error detection (ECC, various fault detection...), measure the chip temperature. That's it, more or less.
