Nimbostratus
MVPa picture equals a thousand words, it would help if you draw the components and the path to clarify what you want.
in principle if the F5 is at the start of the path a health monitor can follow the path if the traffic is send through the hops without any need for interaction. as mentioned your exact situation isn't quite clear to me, but a health monitor in general isn't going to login to a jump host for example. also a proxy server might cause issue, due to authentication requirements for example.
if you require complicated strings depends on your needs. if you are fine with port 80 listening then just a tcp monitor is enough. if you want to be sure a http server is listening then a http monitor is enough. if you want to be sure a specific virtual host is active or a specific response should be returned then you need to configure send and / or receive strings. but all that depends on what you want. and personally i would start simple and see if that is already not enough.