ATC will specify what they want:
be level by intersection zzzzz, that means it is a climb gradient issue.
by level in x minutes, that is a rate issue, although in the end it is actually a gradient requirement, because the time constraint is based on a distance with a crossing traffic.
climb to xxx, 1,500 fpm or greater. Don't think too much, just comply with that.
Now, as for the way of achieving best rate and best angle:
best angle is green dot, but beware! green dot is an IAS and it increases with mach number, so don't select green dot if you are climbing to a high level. you can calculate the green dot at the target level, it is the actual green dot plus one knot per each level above FL 200. There is also a rule of thumb for calculating green dot based on weight.
Don't be brisk, select the speed little by little as you said, 5 by 5 kt is nice. Also, you can use mach instead of IAS. In this manner IAS will be reduced progressively.
Also, if you can meet the constraint without going slower than best rate speed, then don't go slower than that, as it would be less economical to do so. ECON speed at CI zero, or turbulence speed are considered close to best rate speed.
With this considerations you will avoid zooming up like a rocket and then ending up climbing at green dot speed and 100 fpm with two or three levels to go yet.