Hi, April and May are usually good months to avoid much rainfall (and tourists) in South Central AK. You will find a lot of snow in the passes above 2,000 feet or north of Willow, AK on the Parks and Richardson Highways. (Roads plowed and sanded if they are not closed for the winter). Overnight temps will be below freezing and the following are usually closed until May 15th or later depending on seasonal conditions (The southern AK panhandle has different winter closures):
Denali Hwy / Edgerton Highway and Hatcher Pass are usually winter closed till June. go to AK Department of Transportation website for updates: http://511.alaska.gov/alaska511/mappingcomponent/index
Denali National Park - open but only the first and sometimes second campground from the highway and limited visitor center hours. The bus system to go father into the park is not operating till the last week of May. So if you want to see the mountain before then, go to Talkeetna for a view. All of the tourist facilities (aka glitter gulch) north of the Park boundary are closed till mid May.
State Parks - most are gated, some are not but will have the pit toilets and water pumps locked and no trash service or firewood. If they are located snow machine recreation areas, they will be open nearly year around. AK State Parks does not have a park open status website
Chugach National Forest (go to the reservations site for campgrounds america) - Portage Glacier is a great campground area south of Anchorage and is partially open if the snow has melted
Most private tourist attractions are closed till Memorial Day. Public attractions have limited open hours.
Kenai Peninsula: Homer Spit camping is open and exposed to brisk weather conditions but worthwhile for the lack of crowds and awesome scenery, but many seasonal dining facilities closed. Ditto Seward, Hope and Ninilchik. There is usually one or two good clamming openings in mid to late May on the Peninsula
You will encounter lots of frost heaves, melt water and some potholes on the interior highways that time of year. Drive with care and enjoy the views.
FYI: Diesel is available at most gas station in AK statewide -except most Shell stations. Cell phone coverage is available on all of the year around open highways, but not the more remote ones except near towns. Rural towns off the road system often do not have AT&T cell coverage -just the local GCI cell service whichl may discontinue Verizon cell service starting early next year. Verizon is anticipated to start coverage on the main highway system in early January. Check your cell plan many charge roaming fees for data and calls in Canada unless you bump your coverage - AT&T charges $25 for a month to include Canada on my plan or .80 cents a minute if you don't.
The Milepost is worthwhile getting for highway travel information statewide but I found the information on private facilities and when they are open is sometimes not up to date due to weather or publication deadlines. Text and map information on many public campgrounds and some private facilities is vague or limited as they are catering to their nearby private vendors who pay for advertising or placement in the Milepost. The Alaska Marine Highway site has some links to nearby public attractions.