Muroran is not usually a place that first or even second time visitors to Hokkaido will think of, but for those the venture beyond the city itself, the coastline that frames the city culminating in Cape Chikyu has plenty to offer, including some spectacular lookouts and it's a great place to go whale watching.
The best way to explore the area is undoubtedly by car. The coast is several kilometers long and it's not easily accessible by public transportation. If you can rely only on your legs, your best option is to get to Cape Chikyu by bus. You can then hike on the trail that goes up north passing by the nicest spots nearby. Another option is to get here by motorcycle or bicycle, it was even part of stage 5 of the Tour de Hokkaido bicycle race to Sapporo.
When you will arrive at the top viewpoint over the lighthouse, it will take you a few moments before noticing it. The view from there is so wide over the horizon that you can actually see the Earth's curvature. It's impressive. Similar to a mountain summit, the picture of a cape is somehow related to the idea of reaching a goal. That is probably the origin of that feeling of fulfillment and satisfaction this kind of places give. Needless to say, here that sensation it's particularly strong. The constant wind blowing from the ocean, the clouds moving fast in the sky and the amazingly wide view make this place a must-see.
The whole south-eastern coast also offers great spots. Vertical cliffs almost one hundred meters high tower over the ocean while countless seagulls chirp loudly. The bushes covering the top of the shore look like a green wavy sea under the strong winds. Regardless to the weather or season these views are undoubtedly spectacular.
Furthermore, if you are coming from Hakodate by car (the rental rates are way cheaper than the Sapporo area), I personally suggest to skip the highway and follow the provincial road that runs along the ocean. You will save the toll money (more than ¥3000) and you will be rewarded with some great views. The sight of the Mount Komagatake from farther north can turn in some epic scenes in a stormy day.