You might want to consider SFP modules/transceivers integrated into the switches instead of unmanaged converters - SFP modules in managed switches are much better to monitor and troubleshoot.
In your diagram, you connect the converters to the switches using multi-mode fiber. If those switches have SFP slots, just use single-mode transceivers instead and leave out the media converters. Make sure your buying compatible SFP modules, many switch vendors try to lock you in on their "original" modules.
Most media converters are no real bridges (which selectively forward by MAC address) but dual-simplex repeaters - these are capable of full-duplex transmission, yet they forward everything they receive. This doesn't really matter when they're connected to switch ports though.
Having multiple switches or converters in between two normal switches isn't really a problem (technically you can chain as many as you like) but it can make it harder to locate a problem when the link fails. Also, a larger number of devices fails more easily than a smaller number. I'd seriously recommend SFP modules in managed switches.
Twisted pair has a maximum reach of 100m (90m solid-core and 2x 5m stranded cable), so you'd need at least nine switches for regeneration distributed over the distance - that is usually very cumbersome when a problem occurs. With unmanaged switches, you'd have to literally check each switch and the links physically, in person. With managed switches, you'd still have more hardware to monitor and manage than with a single, longer-reach link.
Using single-mode fiber (SMF) and a single pair of 1000BASE-LX tranceivers will provide for better and more reliable service. 100BASE-FX on multi-mode fiber (MMF) would also make the distance, but likely won't be cheaper and MMF won't work for a gigabit upgrade. SMF is good even for 10GBASE-LR and beyond.
If the link is critical you should consider providing some redundancy. With managed switches you could aggregate two 1000BASE-LX links, so the links become redundant. If you'd like redundant switches you can run two links, each terminated on a different pair of switches, and use a spanning-tree protocol (RSTP or MSTP) to resolve the otherwise present bridge loop.