You will need a signal generator which covers the IF frequency and the wavelengths of the set together with a means of measuring the set's output.
You also need some non-metallic trimming tools suitable for the IF transformers and RF coils.
I use a Marconi RF signal generator and an old AVO. The generator has a decent attenuator and the AVO is set to an AC current range and connected across the set's loudspeaker. Using a modulation frequency of around 1kHz produces a clear indication of the set's output. Use sufficient signal to overcome background noise.
1 There are a number of things to check, firstly ensure that the various tuning slugs are moveable and if necessary use a solvent to get rid of any locking agent. Once a dust core has been damaged, often the only recourse is to drill it out and fit a new one so extreme care must be exercised when attempting to turn a slug. Find out the IF frequency of the radio and check on any requirement for stagger tuning (sometimes the case when the IF is around 100kHz). Look for markings on the dial or on the mechanical tuning assembly to tell you where to set the pointer during RF alignment. If the set has to be removed from the case and the dial itself is left behind make sure that you put markings on the set so you can tell what the frequencies are for various dial settings. If the set has short waves try and find out if the oscillator is higher or lower than the dial markings.
2 First set the IF coils to the design frequency. If you have a "wobbulator" or a spectrum analyser it will be more interesting but a simple signal generator will be adequate for all but the most complex communications receivers. Usually one can plug the generator into the aerial socket and tune the receiver away from stations. As alignment progresses back off the generator signal and turn down the volume control. If you find that one of the coils does not peak nicely it may be that the iron dust core has broken away from the screw. This often happens when rust binds the core to the inside of the coil former. Detach the transformer if necessary and retrieve the slug. It may be possible to superglue or araldite it back otherwise you will need to find a replacement. Passing over the transformers several times backwards and forwards will complete the work and hopefully you will be rewarded with a more sensitive receiver.
3 Aligning the RF stages is more complicated. Remember that it is vital that you need to adjust capacitors at the HF end of a band and coil slugs at the LF end. Ganging of tuned circuits is a complex design problem. At every point on the tuning scale the set must run at its peak performance; this despite the fact that the oscillator is tuning a completely different range to the incoming signal and that the various coils are connected to different circuit components having different characteristics. The major design problem is tracking the oscillator so that the dial reads correctly and to this end you may find that the tuning capacitor has different physical sizes for its sections. Often though, the capacitor has equal sections having the same number of vanes, but in series with it is a small padding capacitor. The value of this component is very critical. Also critical, particularly for short wave bands, is the physical layout of the wiring. Incorrect stray capacitance will make the receiver deaf at the HF or LF end of a band.
4 Set the generator to the low end of the medium or long wave bands, say 600kHz and position the receiver dial to 500metres or 600kHz (which is the same thing) and tune the oscillator coil slug for maximum signal. Retune the generator and the receiver to the HF end of the band, say 200metres or 1500kHz and tune the trimming capacitor for maximum signal. Repeat this process a few times and all should be well and the set will have shown a marked improvement. Continue now to align the RF circuits in the same way. The set should now be very lively.
5 Short wave bands are more of a problem. A superheterodyne receiver will always receive at least two generator signals at any given dial setting. The lower the IF the more readily will the receiver pick up two signals. The signals will be spaced apart by twice the IF frequency so that with an IF of 465kHz and with a dial setting of 15,000 kHz or 30metres the signal generator can be set to either the true signal of 15,000kHz or it's image of 15,930kHz. This is because to receive a 30metre signal the oscillator will be set to 15,465kHz. The mathematics of the superhet dictate that signals of oscillator plus or minus the IF can be received. It's the job of the RF stage to both provide gain at the true frequency and rejection of the image. It's of paramount importance then that the RF stage coils are very accurately tracked to the oscillator otherwise the true signal will be rejected and the image enhanced. It's possible given enough patience to do the job well. Component ageing however may mean a lot of fiddling as the oscillator may not track against the dial markings. To some extent one can tweak the outer vanes on the tuning capacitor to line up with dial markings over an entire short wave band but there will always be a compromise. Even if the padding capacitor is within spec and the RF valve input capacity is as the designer assumed, tolerances will come into play to produce variations between perfection and what can actually be achieved. When there is more than one short wave band don't expect to get everything just right. Replacing a frequency changer valve with another manufacturer's equivalent can put circuit stray capacitances miles out with consequent dial errors and anyway very few receivers will provide good image rejection above 20metres so don't expect too much. Basically if anything other than a proper "Communications Receiver" is lively and picks up the main short wave bands at approximately the marked dial settings things are probably as perfect as one can practically achieve.
6 It may take a bit of trial and error to work out whether the signal you're listening to during alignment of a short wave band is the true or the image frequency. If things go from bad to worse during tuning up you probably got the wrong one. The oscillator is sometimes chosen to be high on one band and low on another.