If you are really using the RTM version of vSphere 4.1 without updates and also View 4.6 RTM you have first to upgrade to vSphere 4.1 U3 according to the Interoperability Matrix found here VMware Product Interoperability Matrixes because beginning with View 5.0, vSphere 4.1 RTM is no more supported.
After your vSphere environment is upgraded to 4.1 U3 you can upgrade View to 5.3 as stated in the View 5.2 Upgrade Guide found here http://pubs.vmware.com/view-52/topic/com.vmware.ICbase/PDF/horizon-view-52-upgrades.pdf. After you upgraded View you can then upgrade your vSphere Environment to 5.5.
But here is an advice from my side. Before you upgrade your environment to 5.5 check if all your Hardware (Server, Storage) is on the VMware HCL VMware Compatibility Guide: System Search and with the appropriate Firmware version. If this is ok check if your Backup software is also supported for vSphere 5.5. Let's check out my Backup Compatibility Matrix http://www.vbrain.info/?page_id=1076. Maybe it will help you with this process. Also when using ThinClients check if they are supported with View 5.3.
regards,
Fred