Yes, most hypertension does have a strong genetic component to some extent. In general, hypertension is divided into two sections based on whether the exact underlying mechanism is known or not. Secondary hypertension is relatively rare as far as a cause for elevated blood pressure and can be the result of an underlying hormonal condition (many different hormones can affect blood pressure--cortisol, epinephrine, aldosterone, renin, etc.). Narrowing of particular major arteries such as those that lead to the kidney can also raise blood pressure. These secondary causes for hypertension are usually not considered to have a strong genetic component, but are important to consider as underlying causes because of the potential for "cure".
The other category is primary or "essential" hypertension, which comprises over 90% of patients. The underlying reasons are less well characterized/understood and probably represent the end result of both genetic and environmental interactions. The genetic component is comparitvely weak in the traditional sense suggesting that it might actually be a combination of genes that together contribute to the net effect of raising blood pressure. However, it is also fairly well recognized that our environmental factors (sedentary lifestyle, obesity, salt, diet, etc) can also contribute to either minimize or amplify an underlying genetic risk.
In most cases, we define 'genetic risk' as having a first-degree relative who developed high blood pressure before the age of 55 years (the actual age cut-off is variable). In those cases, most people develop essential hypertension in their 30's-40's, again depending on the environmental components, which can either delay or accelerate onset.