Ashes can be quite alkaline at least for a short time until it leaches away. If your pH is already high I would not put ashes into the soil. If your soil is acidic it probably won't hurt.

Typically it's not a good idea to till uncomposted stuff directly into the soil in spring because breakdown can temporarily deplete nitrogen. Grass clippings are high in nitrogen though so it might be OK. I have not tried this so I'm guessing. I normally use grass clippings mixed with last year's leaves as a summer mulch for the garden. By fall it has broken down and disappeared.

You should really use a mulching mower and recycle those clippings back into the lawn, or mix them with some browns (leaves, straw, sawdust, wood chips, newspaper shreds) and compost them. Your soil will thank you. :-]