Answer:
By making a paternity test by Short tandem repeats, or STRs (microsatellites).
Explanation:
Short tandem repeats, or STRs, are microsatellites consisting of repeated sequences of 2 to 6 base pair length monomers in tandem, and most of them have a high degree of polymorphism. Half of an individual microsatellites are inherited by each parent, and they can be assesed by PCR amplification and further electrophoresis.
In this case, you would try to recover a bone from the remainings of the woman, the Tsar and his wife, since the rest of the latest were in a shallow grave with many different individuals. Preferably, you should manipulate this samples in an ancient DNA facility (aDNA) with proper techniques. You clean the bone samples, drill them, and from internal bone powder make a DNA extraction. After this, you perform a series of PCR targeting a panel of STRs, and electrophoresis. If the woman claim is true, half of her microsatellites should have correspondence with the Tsar, and half with the ones of his wife.
