Many obstacle departures and standard instrument departure procedures have minimum climb requirement or if the minimum climb cannot be maintained takeoff minimums above standard—usually both ceiling and visibility. E.g.
Standard takeoff minimums don’t apply to Part 91 operators but they are a very good idea if you don’t want to hit anything on the way out. For commercial operators standard takeoff minimums are defined by the number of engines. One and two engines: one-mile visibility (1 mile) Three or more engines: one-half mile visibility (1/2 mile). There is no ceiling minimum.
An runway may have alternate takeoff minimums indicated by the inverted T in the header section.
Part 91 pilots are not required to follow the visibility minimums but considering that they were designed to keep airplanes from hitting obstacles that they can’t see on the way out of the airport, it would be foolish not to.
A Part 91 operator is required to meet the climb gradient. That’s what keeps you from hitting something on the way out and being able to safely transition to the enroute system.
Suppose that you can’t meet the climb gradient. Often using a different runway will allow you to depart. Another option that may be available is a Visual Climb Over Airport. If the airport has high enough ceilings, you may be able to climb over the airport until reaching an altitude where you are above the limit of the climb gradient on the departure. Then you can join the departure procedure (assuming that you can maintain the standard 200′ per nm climb). Note that this is an IFR procedure that requires a climb in visual meteorological conditions. As explained in this article, and this example, there may be additional restrictions on visibility.