Allow me a couple of comments based on tearing about 10+ dryers down for work. These suggestions are by no means all-encompassing, and my primary concern would be safety. It goes without saying that modifying a 5kW device is something that should be done VERY carefully. Based on the usual 230V power supply in the US for electric dryers, that's about 25 amperes, I suggest having a fire extinguisher on hand and to have the whole assembly downstream from a manual disconnect switch that is rated for the current you're playing with (i.e. 20+ amps). Do NOT rely on the circuit breaker to protect you.
Unless you have the right background, I would hire an experienced Electrical Engineer (i.e. someone who has designed similar circuits) to guide with the design and a good electrician to implement it. Have them study the schematics that are enclosed with the dryer. They are usually hidden inside the dryer inside a plastic baggie. The manual may mention the location.
Study the main PCB also. The relays that are used there will be a good guide towards the MINIMUM relay capacity you'll have to consider for your board / control system. You may need some very fat traces to carry some of the high currents involved or (better yet) use relays that feature 0.25" quick connect terminals for the load switching (preferable, IMO).
As for modulating heat control, if you want to go down this path I'd suggest the use of a 40A SSR (Triac-based) with a big heat sink - see the spec sheets re: the heat sink needs as a function of current and perhaps mount the heat sink in the path of incoming air to help cool it or add a supplemental box fan to keep it cool. I have used the PID library hosted here
for my reflow oven. Works great.
If your dryer has a separate motor for air flow and drum rotation, I would consider going with a dual-channel SSR, that allows you to modulate the air flow as well as the heater any way you like. I would not mess with the induction motor driving the drum. Keep that on a a simple relay, on/off.
I would also mount multiple DS18B20s around the dryer to shut it off if there is a problem (i.e. SSR heat sink, exit air temperature for the drum, air flow motor, housing around the heaters, etc.). I would also confirm air flow and stop the dryer if there is a problem. The induction motor that usually drives both the fan and the drum may have a centrifugal switch that you could use for that purpose.
Let me re-iterate: Unless you know what you're doing, do not attempt this project by yourself. The voltages and currents involved are sufficient to potentially cause significant physical damage, injury, or even death. Hire experienced help to review what you want to do before implementing it with experienced help. The cost of doing so will be significantly less than the potential trouble you can get yourself or your loved ones into.