It means the person has no alibi, so they are going to be in trouble. They cannot prove they are innocent.
In this case, chemin de croix is the burden of the cross.\
< - Et Pauline? - Desastreux. Elle etait au journal le matin, rentree de Quimper ver trois heures, sortie courir. - Sous la flotte? - Pauline court tout le temps. - Ca va etre un chemin de croix, repeta Louis.>
I just notice it's got a cool double-meaing.
Before that in the text, < ... Ca va etre un chemin de croix pour les alibis. Vous etes pourvu? >
There is also the sense of the stations of the cross, in the amount of troubles one by one that will be faced during the procedure of the investigation.