First check the chain length. Do you still have the old chain? If so, count the links, then count the links in the new chain. If you have a quick link find it and use it as a reference point, otherwise wipe off a link and mark it with a permanent marker pen.
Get the bike in a workstand, trainer or somewhere you can turn the pedals manually with the back wheel off the ground and inspect the drivetrain. As you carefully change gear to larger sprockets does the derailleur look over-extended? Is the derailleur unable to extend enough to allow the chain to move to a larger sprocket. Be really careful doing this. If the chain manages to catch on a larger sprocket than the derailleur allows damage may occur.
If the chain is too short get an extra quick link and splice in a inner link from the leftover chain links and the quick link.
If the chain length is OK, something is binding up which is preventing the shifter from pulling the derailleur inwards to engage the larger sprockets. What you need to do is isolate the derailleur, shifter and cable and find out where the issue lies. First inspect the cable run and looks for anything out of place, make sure the housing is inserted in frame stops, shifter, and derailleur etc.
Detach the cable from the derailleur, put a little tension on the cable (you may need to grip it carefully with a pair of pliers), work the shifter and make sure it can pull the cable freely from highest to lowest gears, If you can feel the cable jamming up as it's pulled in you need to search for where that is happening.
If the cable is OK, make sure the derailleur can move throughout its range, as you pedal the cranks by hand you should be able to push the derailleur in (be very careful doing this and don't get fingers caught in the chain). Make sure it can move from smallest to largest sprocket.
If that all checks out, reattach the cable and work through a derailleur setup procedure to make the limits and indexing are set correctly. This Park Tool article and video is easy to follow and comprehensive.
