じっちゃん means “grandpa”. There are several words for “grandfather” in Japanese. The most formal one is 祖父 which is only used when you talk about your own grandfather in a formal situation. おじいさん and おじいちゃん are commonly used for either someone else’s or your own grandfather. Lastly, じいちゃん and じっちゃん are very casual so you’d better not use them for someone else’s grandfather. Also, when you call your grandfather, you never call him as 祖父 but you can use the rest. You can also use them for an old man.
の is a particle for possession.
名 means “name”. It comes from 名声 or 名誉 meaning “fame” or “reputation” so it’s not just a name of someone, it’s a name of someone who is famous or has a good reputation. So じっちゃんの名 means “the name of my grandpa”.
にかけて means “swearing on” which often goes with nouns indicating human values such as 命 meaning “life” or 信用 meaning “trust” to express strong determination. Also, it is commonly used with te-form.
So the whole sentence in this scene will be じっちゃんの名にかけて俺は必ず貴様を捕まえてみせる meaning “In the name of my grandpa, I will catch you by all means”. But Hajime often says this after the main clause for emphasis and to make it sound catchy.