Do I need to charge VAT when I claim expenses from a client?

Yes. If you were an employee you could make an expense claim without charging VAT but as a supplier to your client, any time* you receive money from a client you make a sale.

All sales need to have VAT added, assuming your business is VAT registered.

*There is an exception to this general principle called 'Disbursements', however it is rare that expense claims can be classed as Disbursements.