Strong odors and airway irritants can be non-allergic triggers of asthma.
First, I would recommend using fewer toxic cleaning products like bleach. Warm water and soap work well for most things. If possible, open windows and run exhaust fans while cleaning. Avoid aerosol sprays.
Wearing any mask while cleaning may help. An N95 mask might work best if you can find one.
You can find more information on cleaning best practices here.