I ran into this issue with with my new Dell XPS 8700. Fortunately before even first booting the hard drive with windows 10 on it I Clonezilled and dd the entire base build so if I got into trouble I could restore to the original factory burn. Newer computers with Windows 8 and 10 use EFI partitioning instead of the old MBR schema. The information where your computer boot data is normally stored in the partition /dev/sda1. Back up that /dev/sda1 and keep it safe. If that partition becomes corrupted the whole system crashes. For me the Grub2 loader, wasn't the issue it was a corrupted EFI partition. No Boot repair is going to fix this problem.
Like you I tried to re partition the windows 10 drive to make room for Ubuntu. Bad mistake. I used the Live CD version of GParted and wound up with multiple partitions also and then when booting up again Windows 10 distroyed the Grub 2 boot loader on update. After 3 tries of restoring the original factory build and then trying to re-partition I used a utility call gdisk to see what is going on with the EFI partition (/dev/sda1).
Unless you are experienced with working with this utility I wouldn't attempt to do a repair with it. I tried and made things worst. Anyway I saw that Windows had rewritten over the /dev/sda1 partition and all data relating to grub2 parameters was lost.
Ok so how did I finally get my partitions resized to what I wanted them to?
After all this installing and uninstalling I ran Dban to clean up the hard drive for there was so much trash on that drive I figured it couldn't hurt. I restored my computer hard drive to factory state inserted the Ubuntu 15.10 disk in and ran through the usual questions for re-installation and used the partitioning tool utility in the install disk to resize my partitions to the size I wanted. (Normally you get to that tool by picking something else on how do you want to install Ubuntu). Then I installed Ubuntu into the new partition manually (refer to the Ubuntu manual if you do not know how to do this). After booting grub came up and I ran windows and ubuntu and all were happy. I tested to see what happened after doing Windows and Ubuntu updates by using gdisk and the efi partition was still in tack in concerns to my /dev/sda1 partition. Never ever touch that partition without backing it up.
So my suggestion is to first see if my scenario is applicable to your computer. Your description is so close to the experience I had it could well be. If your using a legacy MBR/BIOS boot then none of this is applicable. Since the default of most Windows 10 installations require EFI I'm assuming that is what your computer is using.
If your situation seems similar to mine I would highly suggest backing up any data which may be on your hard drive and restoring to close to factory state as possible. My Dell had a restore disk which did a fairly decent job of putting back all the partitions correctly. I only used the clonezilla backups because I wanted to better understand what is actually going on and starting from scratch was the best way to do it.
Load the Live Ubuntu CD disk in. I'd highly suggest using 15.10 + because it understands EFI partitioning much better than its predecessors. I had trouble partitioning with 14.04 LTS. Partition the way you would like to and install Ubuntu into the manually created partition. Be aware to give Windows 10 enough room to work.
Run through the Ubuntu install process and hope for the best.
Some pointers
If you decide to use Clonezilla for a backup, and use images, make sure you save (saveparts) each partition separately. If you make one image of your hard drive Clonezilla screws up the restore because it gets confused between mbr and efi or uefi.
Make sure before doing a Clonezilla backup that your windows 10 is fully updated. There is a dirty bit flag in Windows that Clonezilla picks up and unless you check in the advanced setting to ignore it the program will not copy your windows partition. Best way to avoid this is to make sure Windows is fully updated.
I would suggest for the first backup also do a dd. It's a pain but no matter what it will restore your drive to the original state.
Make sure you use the latest of any software to insure there is a fighting chance the program understands efi. There are plenty of them out there that don't and if that efi partition is not updated trouble will ensue. As mentioned I found the Ubuntu 15.10 Live CD did a pretty decent job on making sure the EFI partition (/dev/sda1) was updated but GParted crashed the hard drive.
- DBan did not run from an EFI partition boot and I had to change the Setup Utility to legacy MBR/Bios boot in order to get that program to work. Use that program with caution because it will completely wipe out your hard drive data! You could skip this step but if your getting to many problems occurring I found this cleared them out for me.
As always proceed with caution and best of luck.
ZV