In the home improvement universe, there's little you can't do. After all, it's your home. That doesn't mean that there aren't some things you may not want to do because they are a lot of trouble and you're unlikely to be satisfied with the results. Your plan to remove laminate from cabinets and paint the pressed wood may be one of those.
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The operative word is "may," because it's also possible that your plan will produce great results, but there's an important condition. The cabinet should be made of medium density fiberboard (MDF), not particleboard. MDF forms a solid core underneath the laminate, and it's a better substrate for painting than particleboard, which is a loose conglomeration of wood chips that are likely to separate and crumble when you pull off the laminate.
Don't Mess With Particleboard
Exposed particleboard is not an attractive cabinet material. It doesn't have a smooth surface, and when you remove the laminate from the cabinets, some of the wood chips that constitute the particleboard are likely to come off with it. You'll be left with a mottled surface that requires the application of a surface coating to give it a smooth appearance, and any such coating you apply prior to painting is in danger of peeling, cracking or moisture damage.
Your plan to peel the laminate from the cabinets is likely the result of the fact that it has already started to peel, and that should tell you the nature of the substrate. If the laminate isn't peeling, lift the edge from one of the doors to get a look at the underlying material. If it's particleboard, save yourself a lot of pointless trouble and either paint over the laminate or consider getting new cabinets.
Look for an MDF Core
Unlike particleboard, MDF is made from compressed fibers, not wood chips, and it's smoother and harder. Painted MDF is a common cabinet material that, unlike particleboard, gives the impression of real wood. It's also solid enough to remain intact when you pull the laminate off. You can recognize MDF underneath a laminate covering by its dense, uniform appearance that lacks the graininess of particleboard.
Once you verify that the cabinet is made from MDF, you're good to go, but getting the laminate off probably won't be easy. The best way to do it is to use a heat gun to soften the adhesive and pull the laminate off in sections, starting with the edging. Some MDF cabinets have a plastic coating, and removing a plastic coating from kitchen cabinets is usually easier than removing a wood laminate. However, both are doable with patience and a steady supply of heat.
Painting MDF Cabinets
After you've removed the laminate, there is bound to be some residual glue. Scrape off as much as you can using a solvent, such as paint thinner or acetone, to soften it. Use the solvent to rub off whatever you can't scrape off. You can also use 120-grit sandpaper, as recommended by Bidvine, for removing glue from MDF cabinets.
Once the MDF is clean and ready for painting, give it a coat of stain blocking primer, which you can apply with a paintbrush. When the primer dries, apply at least two coats of paint, either with a paintbrush or by spraying. Spraying is more trouble, and it requires special equipment, but it gives the best results.