When installing cabinets, the "go-to" method is to "hang" them on a rail that you install first. The idea is that the rail, being much lighter and less bulky, is easier to put in place, get level, and then screw into the studs. Sometimes it's just one rail, sometimes two; depends on the cabinet system. The rail should be securely screwed into every stud along its length, to provide a solid anchor for the cabinet's weight.
Then, the cabinets will have a complimenting hanger hardware that will allow the cabinet to hang on the rail(s). That's enough to support the weight of the cabinets and what they hold, but you should still secure the cabinets to the rail, and thus to the wall, with screws through the backing and/or the back edge of the cabinet side panels. Again, how this works depends on the cabinet system; most cabinets have a pretty flimsy backing, but pre-drilled holes to put screws into the rails behind. Then, you put some trim around the back to hide any gap caused by the rail hardware, and you're done.
Everything Matthew said is accurate, good answer if you're mounting very close to the ceiling. A ledger is a good way to keep the uppers aligned and support the cabinet weight while installing.
There are several other techniques you should know however.
1) Screws holding the weight should be through a framing member or stile, not through a thin backer panel, they should go into a stud or a horizontal nailer installed between studs for this specific purpose. you may have to remove a strip of sheetrock to install a nailer. Do not screw into just sheetrock, even with anchors or your cabinets will be on the floor before you know it. Don't use drywall screws, get cabinet screws that have a built on washer.
2) Adjacent cabinets need to be clamped together and fastened together through the side stiles with counter sunk screws. This locks them together and assures the meeting stiles are flush and stay flush.
3) Always start in a corner if you have cabinets going down two walls. the corner cabinet is your anchor.
4) Carefully check the level of the floor and use embossing material to fill and sags or dips.
5) Dry fit the lowers, before attaching to the wall, shim the low ones, clamp the adjacent stiles together as you did with the uppers. Screw them to the wall, checking level in both directions (length and perpendicular to wall), then screw the stiles together as above.
6) measure precisely the openings required by your appliances. Give yourself no more than 1/4" over the actual appliance dimension. 1/8" is better. 1/16" on either side.
7) Install the lower kick plate last to cover and shims and gaps.
There are a lot of other tricks of the trade, but go slow, pre-plan and have a little extra man power to handle the cabinets.
Google up "This Old House" and watch some of the many videos on this topic.
Best Answer
Adding a wall beside an island seems a major step backwards from current trends of open and airy interior architecture, which have been steadily embraced for decades. The obvious solution would be to add a drop down wall, or even just suspend cabinets from the ceiling, like this:
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(found here).
One idea to help integrate that into the kitchen would be to add beams over the rest of the room:
To do this right requires cutting some holes into the ceiling, tying bracing into the ceiling joists, and then repairing the ceiling. This is a medium sized job for a competent contractor, costing anywhere from $500 to $3000 depending on the scale and specifics of your house. As a do-it-yourself, it is perhaps $50 in materials plus the cabinets and 20 to 60 hours of effort, most of which is drywall work to match the rest of the ceiling.