I've had to do pretty much this at a previous RL job, so I'll go through the issues I can see here and give at least an outline description of what we did, although a) it was about 20 years ago so my memory might be at variance with reality, b) it was on an intrinsically safe system which adds extra components to limit the available power under fault conditions, and c) I wasn't the original designer.
The block-level circuit was a switched current source (stable, reasonably accurate but not to the precision required for measurement) feeding the Kelvin-connected PRT sensor and a high-precision reference resistor (0.01%), with various points fed through protection resistors and a multiplexer to a 24-bit dual-slope integrating ADC. This gave an accuracy of 0.01C in the middle of the range, but only 0.02C (0.013C IIRC) at the high end because of leakage currents acting on the protection resistors, low end fixable as noted below. Using a reference resistor and measuring ratiometrically avoids the need for an accurate and stable current source and relaxes the constraints on the ADC reference so that a normal commercial component will suffice.
I assume the measurement point is remote from the electronics (the sensor is at the end of some cable), because otherwise you're going to have major problems with the electronics being outside its specified temperature range (the normal industrial range is -55 +85C). This fairly well dictates using Kelvin connections (a 4-wire PRT) so that the cable resistance can be eliminated from the measurement - the excitation current is sent down one pair of wires, and the voltage is measured on the other (where cable costs are very high, you can use 3-wire with balanced lengths and compensate for the common wire with some more measurements and software). The basic measurement is to measure the voltage across the sensor and across the reference resistor; being the same current this lets you calculate the PRT resistance and thus calculate the temperature.
Switching the excitation current avoids self-heating whilst allowing an excitation level high enough to give reasonable signal levels; you can choose the excitation current so that the highest sensor circuit resistance gives a voltage near to full-range but still in the linear region, taking into account the resistance of the sensor, reference, connecting cables, temperature variation of these, temperature variation of the current source etc. You could set the excitation current by DAC output (a real DAC, not the PWM lines) and use software to adjust the drive level over the long term to keep the highest ADC reading close to full range - this would avoid loss of resolution at low temperatures (low PRT temperature = low resistance = low ADC reading = fewer bits per degree = reduced accuracy). For the system I had (a fixed current) the power levels were so low that there was negligible self-heating during the measurement period, but if self-heating is an issue you could take multiple readings (at least three) and calculate the t=0 resistance assuming an exponential-asymptotic temperature curve (like V in a CR timing circuit, take measurements at t1,t2,t3 and project back to get V or T at t0; three timed readings are needed to avoid having to know the time constant and final V or T).
Using a single ADC avoids issues of (mis)matching of the ADCs introducing unmeasureable errors; my system had the ADC configured as single ended but you may find that a differential input configuration simplifies matters, however watch for leakage currents and how they vary with input common mode. Using a dual-slope converter you need to use polypropelene or polyethylene capacitors in the ADC circuit to minimise dielectric absorption, these are big and expensive (and also use guard rings on the PCB, and minimise certain PCB trace lengths since the epoxy in FR4 has high dielectric absorption). A delta-sigma converter avoids that but introduces problems with settling time on change of input signal (throw away the first N readings) which extends the measurement time and may allow self-heating to start affecting the readings or prevent timely read (which is why the dual-slope was chosen, with the components available at that time). If there is a gain block available on the input to the ADC it's worth using it to allow the excitation current to be minimised, but don't try to get cute by changing the gain between readings as the gains are never exactly the nominal values, so ADC readings taken with different gains aren't compatible for this purpose.
Another pernicious source of error is unintended thermocouple junctions; even the tin plating on copper wires (or PCB traces) can give this effect. Besides trying to minimise the number of dissimilar metal-metal joints in the signal path, ensure that any you can't avoid are in balanced pairs and isothermal so any effects cancel, and that the signal path is kept as far as reasonable from higher current traces. Be careful of your circuit grounds; have the ADC input side ground (which may be used as a reference for the excitation current source) connected at only one point to an analogue ground (ADC chip and input multiplexer grounds), which is connected at only one point to the system (microprocessor etc) ground which is connected at only one point to the power supply ground input. Another source of error can be input leakage currents; if you have any significant resistance in series with the ADC input (such as the multiplexer 'on' resistance, or a lo-pass filter) check that the voltage drop across this resistance at maximum leakage current is sufficiently small. Also, for this precision, you'll need to ensure that there is very low leakage across the sensor and other parts of the system, such as the reference resistor; anything less than about 10M will have a noticable effect.
When taking a reading, turn on the excitation current, wait a ms or so for it to settle (remember that the sensor cable has inherent capacitance that must be charged to a steady state), do the ADC convertions on all channels on a fixed timing, then re-read all but the last in reverse order on the same timing; perform two more sets of readings if needed to calculate out any self-heating, then turn off the excitation. The nominal time for the set of readings is that of the odd singleton reading (for a dual-slope converter it is the instant that the input sample-and-hold capacitor is disconnected from the inputs), and the pairs of readings should be the same but if they are different, possibly due to self-heating, you can average them to give an equivalent reading at the nominal time. With a 4-wire PRT you have the PRT reading and the reference reading, multiply the reference resistor value by the ratio of these to get the PRT resistance; for 3-wire PRT subtract the reading across the drive wire from the PRT reading first to compensate for the common line. To read multiple PRTs you could either string them in series if the current source has enough compliance and have the input multiplexer with enough channels to select any of the sensors (or the reference resistor), or multiplex the drive - you still need a wide input multiplexer, but the current source compliance requirements are relaxed.
To convert PRT resistance to temperature you could try generating or looking up a formula, but the system I had used the manufacturer R-T data tables and did quadratic interpolation on the three closest data points; this allows easier changing of the sensors used (just put the new table in) or individual calibration by substituting a table of measured values.