It depends exactly what you want checked.
because it is screening you want, I presume that you are subjectily fit and well.
Basic investigation like blood pressure, blood lipid profile ( cholesterol) and glucose, and ecg can be done by your GP quite easily, and for most parts are all that are needed.
Can you exercise without chest pain? A stress test, where you run on a treadmill, is generally done for symptomatic people, but can be done for the worried well who are willing to fork out for it. This can really only be done in a clinic.
Remember, no investigation is without risk. How will you handle an uncertainty in diagnosis where there is no conclusive guidance on how to treat ( a borderline cholesterol, for example).
It is possible to book directly with private hospitals for these, but it is mostly GPs that staff these screening clinics.