Some things they check usually in the lab.
Complete blood counts usually, hemoglobin, hematocrit, MCH etc
Leukocyte profiles. Percentages of lymfocte, monocyte, basofiles, neutrophil,.. and their absolute counts.
Thrombocyte counts to check on coagulation issues.
Iron too usually.
Immunoglobulin issues, deficiencies, reactions. etc.
For diagnosis, I think it's not very reliable to diagnose crohn's disease through blood, CRP, ASCA, Quantiferon for TB etc.
CRP is high in a number of diseases and it doesn't accurately reflect inflammation in the instestine I feel.
ASCA (anti-saccharomyces) is not reliable to diagnose crohn's diseae.
Quantiferon or other tests for TB is usually negative since intestinal TB is rare in the West, but a TB test should be done regardless to rule it out.
Diagnose should be with biopsy with lab results through histology, looking at granuloma and crypt etc to diagnose crohn's disease.