If your incoming, new hardware is using CPUs that are more capable than your existing, out-of-life hardware, you should not incur any downtime for vMotion migrations from old to new hardware provided you're going from like to like architectures (ex., Intel to Intel). Using HCX in this case is way overkill and dramatically complicates what would otherwise be a fairly simple upgrade. Your process should probably go something like:

1. Deploy new compute.
2. Upgrade/deploy vCenter from 5.5 to 6.5
3. Install target version of ESXi and bring to desired patch rev.
4. Configure shared storage per design.
5. With 5.5 and new 6.5 hosts joined to the 6.5 vCenter, migrate workloads from old to new.
6. Decommission old 5.5 compute hardware.
7. With all VMs on 6.5 environment, upgrade VMware Tools.
8. Upgrade virtual hardware level.