You might also consider a brand new spindle.
Here I'm thinking a replacement spindle from Grizzly or one of the other dealers. Of course is you want 5C support you would have to source a larger spindle. Another popular approach is a 5C indexer spindle for a very light lathe.
Of course you can buy lathe spindles as block assemblies that you can bolt to your machine. Unfortunately such spindles cost more than an imported lathe. However you can always look for a used spindle assemblies.
If you wish to DIY it isn't totally impossible you can get good results buy machining any tapers after final assembly. As for your design; for a 5C spindle you probably can do just as well with Angular contact bearings up front and a standard ball bearing at the rear. Ball bearings would allow for higher speeds which if you expect to do a lot of collet work could be advantageous. As for bearings don't get extremely uptight about having the same size at each end. In fact it would be better to retain each bearing separately. Also the front bearings need to be close to the spindle nose as reasonable. The front bearing group would be retained in place in the spindle housing while the rear ball bearing is free to move.
For the DIY approach there is lots of info on the net. Larger spindles and housings are harder to do well. Of course this depends upon how well your shop is equipped. Further the housing is just as import as the spindle itself. Review as many designs as possible until you reach the point of enlightenment.