It has been raining since morning and the weather is gloomy (today).
It's really gloomy today, it's been raining since this morning.
"It's raining since morning and the weather is gloomy." は日常会話で大丈夫です。
文法的に正しくなるように "It has been raining" (=It's been raining) と言った方が良いですが、ネーティブスピーカーは会話でよく "been" をとばすことになります。"today"「今日」とは言うまでもないですが文の最後に付けられます。
このようにも言えます：
It's really gloomy today; it's been raining since morning.
今日は鬱陶しい、朝から（ずっと）雨だった。（今でも雨だ。）
これは英語で自然な言い方です。2つにわけても同じ意味です。
It's really gloomy today. It's been raining since morning.
It has been raining since this morning. It's such a gloomy day.
It's been raining all day. It's kind of dismal out there.
ーIt has been raining since this morning. It's such a gloomy day.
「今朝から雨が降っていて、とても鬱陶しい日だ。」
「朝から雨が降っている」は It has been raining since this morning. と言います。
a gloomy day で「陰鬱な日・どんよりした日」
ーIt's been raining all day. It's kind of dismal out there.
「一日中雨が降っていて、外はちょっと陰鬱だ。」
It has been raining all day. で「一日中雨が降っている」
dismal で「陰気な・陰鬱な・暗い」
もし雨が土砂降りなら It has been pouring... のように言えますし
しとしと降っているなら It has been drizzling... のように言えます。
ご参考まで！
アイルランド 
カナダ 
